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Interfaces in diffusion–absorption processes in nonhomogeneous media
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Original articles
Interfaces in diffusion–absorption processes in nonhomogeneous media Sergey Shmarev a,∗ , Viktor Vdovin b , Alexey Vlasov b a Mathematics Department, University of Oviedo, Oviedo, Spain b Mathematics Department, Siberian Transport University, Novosibirsk, Russia
Received 19 March 2014; received in revised form 18 September 2014; accepted 20 November 2014
Abstract We study the Cauchy problem for the nonlinear parabolic equation
ρ(x)u t = (a(x)φx (u))x − b(x)h(u)
in R × (0, T ]
with nonnegative coefficients ρ(x), a(x) and b(x). It is assumed that φ(0) = 0, φ ′ (s) > 0, φ ′ (s)/s ∈ L 1 (0, δ) for some δ > 0, h(s) ≥ 0 and h(s)/s is nondecreasing for s ≥ 0. The solution of this problem may possess the property of finite speed of propagation of disturbances from the data, which leads to formation of interfaces that bound the support of the solution. It is proved that the behavior of interfaces can be characterized in terms of convergence or divergence of the integrals x
ρ(s)
x0
s x0
dz a(z)
ds,
Jx0 (x) =
b(x) ρ(x)
x x0
0
s
ρ(z) dz a(z)
ds,
b(x) Jx (x), ρ(x) 0
x
ρ(s) ds
x0
as x → ∞ and
ϵ
ds , h(s)
ϵ
ψ(s) ds h(s)
as ϵ → 0 + .
We derive two-sided a priori bounds for the interface location, establish sufficient and necessary conditions for disappearance of interfaces in a finite time (the interface blow-up), and derive the integral equation for the interface. c 2014 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights ⃝ reserved. Keywords: Inhomogeneous diffusion–absorption equation; Blow-up of interfaces; Localized solutions; Lagrangian coordinates
1. Introduction We study the behavior of nonnegative weak solutions of the degenerate parabolic equation ρ(x)u t = (a(x)φx (u))x − b(x)h(u).
(1.1)
∗ Corresponding author.
E-mail addresses: [email protected] (S. Shmarev), [email protected] (V. Vdovin), [email protected] (A. Vlasov). http://dx.doi.org/10.1016/j.matcom.2014.11.004 c 2014 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights 0378-4754/⃝ reserved.
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The coefficients ρ(x), a(x), b(x) are given functions, a(x) and ρ(x) are strictly positive on every compact subset of R \ {0}, b(x) is nonnegative in R. The functions φ(s), h(s) : R+ → R+ are assumed to satisfy the conditions φ ′ (s) φ(0) = 0, φ ′ (s) > 0 for a.a. s > 0, ds < ∞, s 0+ h(s) ≥ 0,
h(s) is nondecreasing s
for s ≥ 0.
(1.2)
(1.3) (1.4)
Eq. (1.1) appears in the mathematical description of various physical phenomena such as the heat propagation in inhomogeneous medium [21,30], thermal evolution in heated plasma [22], joint motion of salt and fresh water [12,13]. This equation generalizes the porous medium equation u t = (u m )x x ,
m > 1.
(1.5)
Unlike the linear heat equation, the solutions of (1.5) possess the property of finite speed of propagation of disturbances from the data, which leads to appearance of the interfaces. The interfaces are plain curves x = ζ (t) ≡ sup{x ∈ R : u(x, t) > 0},
x = ξ(t) ≡ inf{x ∈ R : u(x, t) > 0},
that separate the regions where the solution is positive from those where it equals zero. It is known that in the solutions of (1.5) the interfaces propagate with finite speed, never stop, and tend to infinity as t → ∞, see, e.g., [17,35]. This type of behavior is typical for the solutions of Eq. (1.1) with constant positive coefficients ρ, a, b, which can be regarded as a model of the process of slow diffusion (condition (1.3)) with weak absorption (condition (1.4)). The situation changes if we consider the inhomogeneous counterpart of Eq. (1.1). Let us consider the Cauchy problem for the equation ρ(x)u t = (u m )x x in R × (0, T ), (1.6) u(x, 0) = u 0 (x) in R, supp u 0 = (−l, l), 0 < l < ∞, assuming that the coefficient ρ(x) is strictly positive on every compact subset of R. It was discovered in [18] that problem (1.6) admits solutions whose interfaces vanish in a finite time, that is, the initially bounded set supp u(x, ·) becomes infinite at a finite moment t = T ∗ . A characterization of this effect in terms of the properties of ρ(x) is given in [21,30]. It is shown that if ρ(x) and u 0 (x) are even functions, then the following alternative holds: ∞ <∞ ⇒ supp u(x, t) = R at a finite instant T ∗ (the so-called interface blow-up), xρ(x)d x =∞ ⇒ supp u(x, t) remains finite for all t > 0. 1 Less is known about the phenomenon of the interface blow-up in the multidimensional case. It is proved in [12] that for the solutions of the Cauchy problem for the equation ρ(|x|)u t = ∆u m ,
m > 1,
(1.7)
in R2 × (0, T ] the set supp u(x, t) becomes unbounded in a finite time if ∞ s ρ(s) ln s ds < ∞ 1
and remains bounded for all t > 0 if ∞ ln s min ξ 2 ρ(ξ ) ds = ∞. s ξ ∈(0,s) 1 In the case n ≥ 3 the effect of the interface disappearance in solutions of (1.7) is studied in [34]. It is shown that the interfaces in solutions of Eq. (1.7) with ρ(x) ∼ x −λ as x → ∞ disappear in a finite time if λ > n(m−1)+2 , and remain m
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bounded for all finite t > 0 if λ < n(m−1)+2 . Paper [34] deals with the doubly nonlinear equations of the form m ρ(|x|)u t = div |u|m−1 |∇u|λ−2 ∇u , u t = div a(|x|)|u|m−1 |∇u|λ−2 ∇u , which contain (1.7) as a partial case. The analysis is based on the study of the local energy estimates in weighted Sobolev spaces. The behavior of interfaces and the possibility of the interface blow-up in solutions of the diffusion–absorption equation ρ(x)u t = (u m )x x − cu p with m > 1, p > 0 and c = const > 0 is studied in papers [23,27]. It is shown that the interfaces remain bounded p−1) as for all t > 0 if either p ∈ (0, m], or p > m and ρ(x) ∼ |x|−k as x → ∞ with k ≤ k ∗ = 2(p−m . If p > m and k > k ∗ , then the interfaces disappear in a finite time. The behavior of interfaces in solutions of the Cauchy problem for the diffusion–absorption equation with variable absorption, u t = ∆u m − b(x)u p
in Rn × (0, ∞],
is studied in paper [26]. It is shown that for p ≥ m the support of the solution covers the whole of Rn as t → ∞ and, conversely, is contained in a finite ball for all t > 0 (localization of the solution) if p ∈ [1, m) and b(x) ∼ |x|−k as |x| → ∞ with k ≥ 2. For k > 2 and p ∈ [1, m) the solution is localized if the initial datum is appropriately small. The aim of present work is to describe the behavior of interfaces in solutions of Eq. (1.1) in terms of convergence and divergence of the integrals s x s x dz b(x) b(x) x ρ(z) G x0 (x) = ρ(s) ds , Jx0 (x) = dz Jx (x), ds ρ(s) ds (1.8) ρ(x) x0 ρ(x) 0 x0 x0 a(z) 0 a(z) x0 as x → ∞ and the behavior of the integrals µ s ′ µ 1 φ (z) ds , dz ds h(s) h(s) z ϵ 0 ϵ
as ϵ → 0 + .
(1.9)
The parameter µ > 0 is defined in condition (2.3). We prove that the behavior of interfaces in solutions of the inhomogeneous equation (1.1) with finitely supported initial data depends only on the nonlinear structure of the equation, which is described in terms of convergence of the integrals in (1.8), (1.9). By now there exist exhaustive studies of the property of finite speed of propagation from the data and the behavior of supports in solutions of homogeneous equations of the type diffusion–absorption with constant a and ρ and b- see, e.g., [9,10] for pertinent results and a review of the literature. We refer also to [1,2], for an analysis of the short-time behavior of interfaces in solutions of equations with power nonlinearities φ(s) = s m and h(s) = s β with m, β > 0. Notice also that since the radially symmetrical solutions of the multidimensional equation ρ(|x|)u t = div (a(|x|)∇φ(u)) − b(|x|)h(u)
in Rn × (0, T ]
(1.10)
satisfy a one-dimensional equation of the type (1.1), the results of the present work may be applied, by comparison, to solutions of certain equations of the type (1.10). 2. Assumptions and results Let us consider the Cauchy problem ρ(x)u t = (a(x) u ψx (u))x − b(x)h(u) u(x, 0) = u 0 (x) in R
in ST = R × (0, T ],
(2.1)
with the initial function u 0 (x) > 0
for x ∈ (−l, l), l > 0,
u 0 (x) = 0
in R \ (−l, l)
(2.2)
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and the coefficients ρ(x), a(x) and b(x) ≥ 0 satisfying (1.2). The function ψ(·) is defined by the relation s ′ φ (z) ψ(s) = dz, z 0 so that s ψ ′ (s) = φ ′ (s), ψ(0) = 0, and Eq. (2.1) is equivalent to (1.1). Throughout the paper we assume that the data of problem (2.1) satisfy (1.2) and the following conditions: ρ, a, b, u 0 are even continuous functions in R, is nonincreasing for x > 0, u0 ∞ ρ(x)u 0 (x) d x = M < ∞, 0 µ = sup u < ∞.
(2.3)
0
R
Definition 2.1. A function u(x, t) is called weak solution of problem (2.1) in ST if 1. u is nonnegative, continuous and uniformly bounded in ST , ρu ∈ L ∞ (R) for all t ∈ [0, T ], 2. √a u ψx (u) ∈ L 2 (S ), T 3. for every smooth test-function Φ, vanishing for all sufficiently large |x| t=T = 0. I (u, Φ) ≡ ρuΦ d x [ρu Φt − aΦx u ψx (u) − Φbh(u)] d xdt − ST
R
(2.4)
t=0
If in the conditions of Definition 2.1 I (u, Φ) ≤ 0 (I (u, Φ) ≥ 0) for every nonnegative test-function Φ, then the function u is called supersolution (subsolution) of problem (2.1). Under suitable assumptions on the data, existence of solutions to problem (2.1) can be proved by known methods— see, e.g., [8,12,19,23,28]. The cited papers use different notions of weak solution, but in every case the solution is constructed as the limit of a sequence of solutions to the problems posed in expanding cylinders with zero Dirichlet conditions on the lateral boundary. Such a procedure leads to the minimal solution of the Cauchy problem. It is noteworthy that in the case when ρ(x) is rapidly increasing as x → ∞, global in time solutions to problem (2.1) may be not unique. We refer to [7,8,19] for more information on this issue. To formulate the a priori results about the interface behavior we assume that the Cauchy problem (2.1) admits a local in time solution which possesses the property of finite speed of propagation of disturbances from the initial data. Namely, we claim that the following assertion holds. Let conditions (1.2)–(1.4), (2.2), (2.3) be fulfilled. Then there exists T ∗ (u 0 ) > 0 such that problem (2.1) has in the strip ST ∗ (u 0 ) a solution u(x, t) that possesses the properties: ∀ τ ∈ (0, T ∗ (u 0 )] ∥u(·, τ )∥∞,R ≤ µ = sup u 0 ,
(2.5)
R
ζ (t) = sup{x ∈ R : u(x, t) > 0} ≤ L
on [0, T ∗ (u 0 )], L = const > 0.
All further considerations will be confined to the time interval [0, T ] with T = sup{T ∗ (u 0 ) > 0 : (2.5) is fulfilled}. Properties (2.5) are typical for solutions of parabolic equations which display the finite speed of propagation and obey the maximum principle. For the sake of completeness of presentation, in Section 3.1 we outline a proof of (2.5) for subclasses of Eq. (1.10). The main results of this work are the following. Theorem 2.1 (The Interface Equation). Let conditions (1.2)–(1.4), (2.2), (2.3) and (2.5) be fulfilled. Then for every t ∈ [0, T ∗ (u 0 )) there exists the limit t a(x) lim ψx (u(x, τ )) dτ x→ζ (t) 0 ρ(x)
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and the following interface equations hold: t a(x) ζ (t) = ζ (0) − lim ψx (u(x, τ )) dτ. x→ζ (t) 0 ρ(x)
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(2.6)
Since the solution of problem (2.1) is nonnegative, it follows from the interface equation (2.6) that ζ (t) is a monotone increasing function. Accept the notations l µ ρ(x) ds K = d x, Hµ (z) ≡ (2.7) a(x) h(s) 0 z with the constants l > 0 and µ > 0 from conditions (2.2) and (2.3). Theorem 2.2 (Localized Solutions). Let the conditions of Theorem 2.1 be fulfilled. Assume that h(s) > 0 for s > 0 and for all x ≥ 0, b(x) ≥ λ(x) > 0 with a nonincreasing function λ(x) ρ(x) and introduce the function R(σ, t) ≡ λ(σ ) [Jl (σ ) − K (ζ (t) − l)] −
tλ(σ )
0
ψ Hµ−1 (s) ds
with Jl (x) defined in (1.8) and the constant K from (2.7). The following assertions hold. (a) Problem (2.1)–(2.2) has a solution satisfying the inequality ∀ t ∈ [0, T )
R(ζ (t), t) ≤ 0.
(b) If the function λ(s) [Jl (s) − K (s − l)] is monotone increasing as s → ∞ and ∀t >0
sup {s > 0|R(s, t) < 0} < ∞,
(2.8)
then T = ∞, i.e., ζ (t) → ∞ only as t → ∞. (c) If lim sup λ(s) [Jl (s) − K (s − l)] = ∞ and s→∞
0
∞
ψ Hµ−1 (s) ds < ∞,
then T = ∞ and the solution is localized, i.e., there exists a finite constant L > 0 such that ζ (t) ≤ L for all t > 0. Theorem 2.3 (Finite Speed of Propagation). Let the conditions of Theorem 2.1 be fulfilled. If b(x)h(u) ≡ 0, then problem (2.1)–(2.2) admits a solution such that ζ (t) → ∞ only as t → ∞, i.e., T = ∞, and t ∀ t > 0 Jl (ζ (t)) ≤ K (ζ (t) − l) + ψ (u(0, τ )) dτ (2.9) 0
with the constants l and K defined in (2.2) and (2.7). The next assertion establishes lower bounds for the interface location and gives sufficient conditions for the interface blow-up. Let us fix a point x0 ∈ (0, l) ≡ supp u 0 ∩ {x > 0} such that x0 M ρ(x)u 0 (x) d x = (2.10) 2 0 with the constant M from (2.3). For the initial data u 0 satisfying (2.3) such a choice is always possible. Theorem 2.4 (Blow-up of Interfaces). Let the conditions of Theorem 2.1 be fulfilled, and, additionally, ∞ ρ(x) d x = D < ∞. 0
(2.11)
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(a) If h(s) > 0 for s > 0 and b(x) = λρ(x) with a positive constant λ, then problem (2.1)–(2.2) has a solution u(x, t) such that ζ (t) s dz ∗ ∀ t ∈ [0, T (u 0 )) G x0 (ζ (t)) ≡ ρ(s) ds x0 x0 a(z) λt 1 M −1 ≥ F(t) ≡ ψ (2.12) H (s) ds. λ 0 2D µ (b) If G x0 (∞) < ∞ and lim supt→∞ F(t) = ∞, then the interface disappears in a finite time, i.e., ζ (t) → ∞ as t ↗ T. (c) If b(x)h(s) ≡ 0 and G x0 (∞) = ∞, then the lower bound for the interface is given by the inequality M . ∀ t ∈ [0, T ] G x0 (ζ (t)) ≥ t ψ 2D (d) If b(x)h(s) ≡ 0 and G x0 (∞) < ∞, then the interface disappears at an instant T ≤
G x0 (∞) . ψ(M/2D)
The proofs of Theorems 2.1–2.4 are based on the nonlocal coordinate transformation that renders the interface stationary. This change of independent variables, frequently used in continuum mechanics, is explained in Section 4. 3. Examples Let us illustrate the assertions of Theorems 2.1–2.4 by the examples of model equations of the type (1.1). In these examples the integrals G x0 (x) and Jl (x) can be calculated explicitly, which provides a possibility to compare the above results with some of the already known results obtained with other methods. 3.1. Local in time solvability Let us consider the Cauchy problem for the equation ρ(x)u t = ∆ u m − b(x)u p
in ST = Rn × (0, T ], m > 1, p ≥ 1,
(3.1)
with a nonnegative initial function u 0 such that supp u 0 ⊂ B1 (0). Assume that the coefficients a(x) and ρ(x) are strictly positive on every compact subset of Rn , but may vanish as |x| → ∞. Eq. (3.1) is of the type (1.10), or (1.1) in the special case of radially symmetric coefficients ρ and b. A solution to this problem can be constructed in a number of ways. For example, it can be done by adapting the arguments of [4,5]. Fix some r > 1 and consider the Dirichlet problem vt = ∆ A(x)v m − B(x)v p in Q r,T = Br (0) × (0, T ], v = 0 on the lateral boundary of Q r,T , (3.2) v(x, 0) = v0 ≡ ρ(x)u 0 (x) in Br (0). Here T > 0 is to be defined and the coefficients have the form A(x) = ρ −m (x),
B(x) = b(x)ρ − p (x).
Problem (3.2) follows from (3.1) after the substitution v = ρ u. This problem is a partial case of the problem considered in [4,5]. Since ρ is strictly positive in Br (0), then A(x), B(x) ∈ C 0 (Br (0)), A(x) is separated away from zero and infinity in Br (0), and B(x) ≥ 0 is bounded. By [4,5], for every nonnegative initial function u 0 ∈ L ∞ (Br (0)) and every T > 0 problem (3.2) has in the cylinder Q r,T a unique nonnegative solution v(x, t) such that ∥v∥ L ∞ (Qr,T ) ≤ ∥v0 ∥ L ∞ (Br (0)) ≡ ∥ρ u 0 ∥ L ∞ (Br (0)) ,
∥∇v m ∥22,Qr,T ≤
1 ∥v0 ∥22,Br (0) . 2
(3.3)
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The solutions of problem (3.2) possess the property of finite speed of propagation of disturbances from the data—see [4], [3, Ch. 3, Th. 2.1]: if v0 (x) ≡ 0 in a ball Bs0 (x0 ) ⊂ Br (0), then v(x, t) ≡ 0 in the co-centric smaller ball Bs(t) (x0 ) of radius 1 1 s(t) = s0ν − C t γ ν for t ≤ s0ν /C ν . The positive parameters ν, γ , C depend on m, p, the norm ∥v0 ∥(m+1)/m,Bs0 (x0 ) and the quotient sup ρ(x)/ inf ρ(x).
Bs0 (x0 )
Bs0 (x0 )
In particular, C ↗ ∞ as inf{ρ(x)|x ∈ Bs0 (0)} ↘ 0. It follows that for every big r > 0 and the sufficiently small T the support of the solution does not reach the lateral boundary of the cylinder Q r,T . Continuing the solution by zero from Q r,T to the rest of the strip ST , we obtain a solution of the Cauchy problem. The first estimate in (3.3) allows us to continue the solution to the strip Rn × (T, T1 ]. This process is continued until the moment when either the support Ti = ∞. In case that the initial function u 0 and the coefficients ρ, a, b of the solution becomes unbounded, or are radially symmetric, the constructed solution of Eq. (3.1) is also radially symmetric and satisfies the conditions of Definition 2.1 in the strip ST . For the equation u t = ∆φ(u) with φ(s) satisfying conditions (1.3), existence of radially symmetric solutions satisfying (2.5) is proved in [24, Ch. 3, Sec. 25]. We refer also to the papers [16,15,25,20,29] for a discussion of solvability of the equation ρ(x)u t = ∆φ(u). The weighted porous medium equation ρν u t = div ρµ ∇u m (3.4) with m > 1 and positive functions ρν , ρµ was considered in [11]. 3.2. The porous medium equation Let us consider the Cauchy problem for the porous medium equation u t = ∆u m
in Rn × R+ , m > 1.
(3.5)
This equation admits the class of explicit self-similar solutions (the Barenblatt solutions) 1/(m−1) x U (x, t) = (1 + t)−α A − B|ξ |2 , ξ= , + (1 + t)β
(3.6)
with the parameters α=
1 , m − 1 + 2/n
β=
α , n
B=
m−1 β, m
A = const.
For these solutions U m−1 (0, t) = A(1 + t)−α(m−1) . The function U is a solution of the radially symmetrical version of Eq. (3.5) r n−1 Ut = r n−1 (U m )r , r = |x| > 0, r
which coincides with (2.1) if we choose ρ(r ) ≡ a(r ) = r n−1 , h ≡ 0. By Theorem 2.3, the interface of the self-similar solution obeys the estimate 2 t m 1 ζ (t) − A/B ≤ A/B ζ (t) − A/B + U m−1 (0, τ ) dτ. 2 0 m−1 It follows that 2 ζ (t) − 2 A/B ≤ A/B + 2 0
t
m U m−1 (0, τ ) dτ ∼ (1 + t)2β m−1
as t → ∞.
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This example shows that estimate (2.9) is sharp because the interface of the self-similar solution (3.6) is the plane curve defined by the explicit formula r = ζ (t) ≡ A/B(1 + t)β , r = |x|. 3.3. The porous medium equation with variable density In the case of the Cauchy problem for the equation ρ(x)u t = ∆ φ(u)
in Rn × R+
(3.7)
with φ(·) and ρ(·) satisfying conditions (1.2) and (1.3) Theorem 2.4 gives the following conditions for the interface blow-up: ∞ xρ(x) d x < ∞ if n = 1, 1 ∞ the interfaces disappear x ln x ρ(x) d x < ∞ if n = 2, ⇒ in a finite time 1 ∞ x n−1 ρ(x) d x < ∞ if n ≥ 3 1
(compare with [12,21,30]). Under various conditions on the density ρ(x), the long time behavior of solutions of Eq. (3.7) was studied in [16,15,25,20,29]. 3.4. Localized solutions of the diffusion–absorption equation Let us consider the diffusion–absorption equation with power nonlinearities ρ(|x|)u t = ∆u m − b(|x|)u p
in Rn × R+ ,
(3.8)
assuming that m > 1, p > 0, ρ(|x|) = (1 + |x|)−α
and
b(|x|) = (1 + |x|)−β ,
α, β ≥ 0.
(3.9)
For Eq. (3.8) under assumptions (3.9) b(r ) ≥ λ(r ) ≡ (1 + r )α−γ ρ(r )
with γ = max{β, α}.
Let us assume, for the sake of definiteness, that the support of the initial function is contained in the ball B1 (0). A straightforward calculation gives: for 0 < l < 1 and µ = supR u 0 x α−γ +1 if α > 1, 1+x 2−γ as x → ∞ λ(x)[Jl (x) − K (x − l)] ≥ C x ln if α = 1, 1+l 2−γ x if α ∈ [0, 1), with C ≡ C(α, γ ) and s
0
p−m s p−1 if p > m, ψ Hµ−1 (−s) ds ∼ ln(1 + s) if p = m, const if p < m.
It follows that the solutions of (3.8) are localized if either α ∈ [0, 1] and β < 2, p < m and or α > 1 and β < α + 1 (compare with [23,26,27]).
(3.10)
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3.5. Disappearance of interfaces in diffusion–absorption equations Let us notice first that since the solutions of Eq. (3.7) with φ(s) = s m , m > 1, are supersolutions of Eq. (3.8), the inequality G x0 (∞) < ∞ is necessary for the blow-up of interfaces in solutions of (3.8). Let condition (3.9) be fulfilled and b(x) = λρ(x). By Theorem 2.4 the support of the solution becomes unbounded in a finite time if ∞ s dz < ∞, p ≥ m and G x0 (∞) = s n−1 (1 + s)−α ds n−1 x0 x0 z that is, if α >
2 n
for n = 1, 2, for n ≥ 3.
Condition (2.11) is fulfilled if in condition (3.9) α > 1. Comparison of solutions of the equations ρ(|x|)u t = ∆u m − λρ(|x|)u p
and
ρ(|x|)u t = ∆u m − b(x)u p
shows that in the solutions of the latter the interfaces disappear in a finite time if 2 for n = 1, 2, p ≥ m and β ≥ α > n for n ≥ 3. 3.6. Eventual positivity Let in Eq. (3.8) p ≥ m, b(x) = λρ(|x|) with some λ = const > 0 and let b, ρ are of the form (3.9). Let us assume, for the sake of definiteness, that n ≥ 3, G x0 (∞) = ∞, and claim that condition (2.11) is fulfilled: 1<α ≤ n
if n ≥ 3.
Under these assumptions the interfaces in solutions of (3.8) cannot display the blow-up behavior and estimate (2.12) holds. Moreover, by virtue of (2.6) the function ζ (t) is monotone nondecreasing. Plugging (3.10) into (2.12), we estimate ζ (t) from below: p−m ζ n−α (t) − x0n−α if n > α, p−1 t if p > m, 1 + ζ (t) C0 ≤ G x0 (ζ (t)) ≤ C1 if n = α ln ln(1 + t) if p = m 1 + x0 with finite constants C0 = C0 ( p, m) and C1 = C1 (α, n, x0 ). 4. The change of independent variables. Lagrangian formulation In this section we introduce the change of independent variables which allows us to reduce free boundary problems for parabolic equations to equivalent problems posed in a rectangular time-independent domain. Let us interpret the Cauchy problem ρ(x)u t = (a(x)φx (u))x − b(x)h(u) in ST , (4.1) u(x, 0) = u 0 (x) in R, supp u 0 = (−l, l), l < ∞, as the mathematical description of the one-dimensional motion of a fluid with the density d(x, t) = ρ(x)u(x, t)
and the velocity v = −
a(x) ψx (u). ρ(x)
Eq. (4.1) expresses the mass balance law in the Euler description, dt + (d v)x = −b(x)h(u), where the term −b(x)h(u) represents the density of the mass forces. The initial condition in (4.1) defines the initial distribution of density: d(x, 0) = ρ(x)u 0 (x) in R.
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There are two methods of description of motions of continua. In the Euler method the characteristics of motion are considered as functions of the time t and a coordinate system not connected with the medium itself. In the method of Lagrange the main quantities characterizing the motion are considered as functions of the time t and the initial positions of the particles. The latter method is of special convenience if the domain occupied by the moving medium is a priori unknown and has to be defined together with the other characteristics of motion. If the medium is constituted by the same particles at each instant, on the plane of Lagrangian coordinates the flow domain is time-independent. From the mathematical point of view, the existence of these two counterpart descriptions of the same mechanical process furnishes a nonlocal coordinate transformation that renders the free boundary stationary (see [24] for more information on this issue). This change of independent variables was already used to study the interface regularity in solutions of problem (1.6) [14]. In the present work we follow the method of introduction of Lagrangian coordinate proposed in [32,33] for the study of interfaces in processes without mass conservation. A local version of the system of Lagrangian coordinates was used in [31,6] for the study of free boundary problems without mass conservation in several space dimensions. Let us denote by y = y(ξ, t) the position of the fluid particle which was initially located at the point ξ ∈ [−l, l]. By ρ(y)U (ξ, t) we denote the density at this particle, so that U (ξ, t) = u[y(ξ, t), t]. We consider the fluid motion governed by the following conditions. (1) The trajectory equation: y = v(y, t) ≡ − a(y) ψ (U (ξ, t)) for ξ ∈ [−l, l], t > 0, y t ρ(y) y(ξ, 0) = ξ ∈ [−l, l]. (2) The mass balance law: the mass of every moving fluid volume ω(t) constituted by the same particles changes with time in a prescribed way. Let us take a fluid volume ω(0), label each of the particles by its initial position ξ , and define the mapping ω(0) ∋ ξ → x = y(ξ, t) ∈ ω(t),
t > 0.
The volume ω(t) is constituted by the same particle at each instant t and its mass is given by the formula m(t) = ρ[y(ξ, t)]u[y(ξ, t), t] dy(ξ, t). ω(t)
Let us assume that the mass m(t) of ω(t) changes with time according to the rule dm(t) b(y)h(u) dy. =− dt ω(t) Formally passing to Lagrangian coordinate ξ , we have: dm(t) d = ρ(y)u(y, t)yξ dξ dt dt ω(0) d = ρ(y)u(y, t)yξ dξ = − b(y)h(u)yξ dξ. ω(0) dt ω(0) Since the set ω(0) is arbitrary, we conclude that d ρ(y)u(y, t)yξ = −b(y)h(u)yξ dt
in [−l, l] × [0, T ].
Dividing the both parts of this equality by ρ(y)u(y, t)yξ and integrating in t, we arrive at the mass balance law in Lagrangian setting: t b(y) h(u) ρ(y)u(y, t)yξ = ρ(ξ )u 0 (ξ ) exp − dτ in [−l, l] × [0, T ]. (4.2) 0 ρ(y) u
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It is convenient to introduce now the mass Lagrangian coordinate η, related to the geometrical Lagrangian coordinate ξ by the formula ξ η= ρ(s) u 0 (s)ds − M : [−l, l] → [−M, M]. (4.3) −l
Using the chain rule, it is easy to calculate that 1 1 Dξ = Dη Dξ η yξ yξ t b(y) h(u) ρ(y)u exp dt ρ(ξ )u 0 (ξ )Dη = ρ(ξ )u 0 (ξ ) 0 ρ(y) u = ρ(Y ) A(η, t) U Dη
Dy =
with t
A(η, t) = exp 0
b(Y ) h(U ) dt . ρ(Y ) U
(4.4)
Gathering (4.2) with (4.3) and considering the trajectory y = Y (η, t) and the density u = U (η, t) as functions of the mass coordinate η, we arrive at the following problem: it is requested to find functions (U, Y ) satisfying the system of equations Yt + a(Y ) A U ψη (U ) = 0, ρ(Y ) Yη A U = 1 in Q T = (−M, M) × (0, T ), (4.5) U (−M, t) = U (M, t) = 0, U (η, 0) = u 0 (ξ(η + M)), Y (η, 0) = ξ(η) in [−M, M]. Such a reduction of the free boundary problem (4.1) to problem (4.5) is based on a formal resemblance between the Cauchy problem (4.1) and the mass balance law in the Euler description. Let us now give a rigorous justification of this change of independent variables. Definition 4.1. A pair of functions (Y, U ) is said to be a solution of problem (4.5) in Q T if 1. U, Y ∈ C 0 (Q T ), U ≥ 0 in Q T , √ 2. a(Y )ρ(Y ) U A U ψη (U ) ∈ L 2 (Q T ) and Yt + a(Y )U A ψη (U ) = 0,
ρ(Y ) U A Yη = 1
a.e. in Q T .
(4.6)
Theorem 4.1. Problems (4.1) and (4.5) are equivalent in the following sense: (a) if condition (2.5) is fulfilled, then problem (4.5) has a solution in the sense of Definition 4.1 in the rectangle Q T ∗ (u 0 ) ; (b) if problem (4.5) has a solution in the sense of Definition 4.1 in a rectangle Q T , then problem (4.1) has a solution in the sense of Definition 2.1 in the strip ST and ζ (t) < ∞ on [0, T ]. Proof. (a) Let u(x, t) be a bounded nonnegative weak solution of problem (4.1) in a strip ST ∗ (u 0 ) . Since the data of problem (4.1) are even in x, so is the solution u(x, t). Let us introduce the new independent variable η and the functions Y (η, t), U (η, t) by the relations η Y (η,t) ds = ρ(s)u(s, t) ds, (4.7) 0 0 A(s, t) x = Y (η, t), U (η, t) = u(x, t) with A defined in (4.4). The function Y (η, t) is monotone increasing in η for every fixed t and the transformation η → x = Y (η, t) is locally invertible in a neighborhood of every point where u(x, t) > 0. Denote by ΣT ∗ (u 0 )
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= (−M(t), M(t)) × (0, T ∗ (u 0 )) the image of the set supp u ≡ (−ζ (t), ζ (t)) × (0, T ∗ (u 0 )) under the mapping x → η. By the definition of η and Y (η, t) M(t) ζ (t) ds = ρ(s)u(s, t) ds. A(s, t) 0 0 By the assumption ζ (t) ≤ L on [0, T ∗ (u 0 )]. Choosing in (2.4) a smooth test-function Φ such that 1 if |x| ≤ L + 1, Φ(x, t) = 0 if |x| ≥ L + 2 we arrive at the equality t=T ∗ (u 0 ) =− ρ u d x t=0
R
b(x)h(u(x, t)) d x dt. ST ∗ (u 0 )
The same equality is true for every strip ST with T ≤ T ∗ (u 0 ). It follows that for every t, t + ∆t ∈ [0, T ∗ (u 0 )] τ =t+∆t ζ (τ ) t+∆t ζ (τ ) 1 1 =− ρ u dx b(x)h(u(x, τ )) d x dτ. ∆t 0 ∆t t 0 τ =t
Letting ∆t → 0, we obtain ζ (t) ζ (t) d ρ u dx = − b(x)h(u(x, t)) d x. dt 0 0 The change of variable x = Y (η, t) transforms this equality into the following one: M(t) M(t) M(t) dη d b(Y ) h(u(Y, t)) 1 1 =− dη ≡ dη. dt A(η, t) ρ(Y ) u(Y, t) A A(η, t) t 0 0 0 It follows that M ′ (t) ≡ 0, that is, ΣT ∗ (u 0 ) = Q T ≡ (−M, M) × (0, T ∗ (u 0 )) with l M= ρ u 0 d x. 0
Given a smooth function Φ(x, t), let us define its “double” f (η, t) ≡ Φ(Y (η, t), t) = Φ(x, t)
for (x, t) ∈ supp u and (η, t) ∈ Q T ∗ (u 0 ) .
It is easy to calculate that dη = ρ(Y ) U A dY, ρ(Y ) U A Yη = 1, Dx = ρ(Y ) A U Dη , Φx (x, t)|x=Y (η,t) = ρ(Y ) U A f η , Φt (x, t)|x=Y (η,t) = f t − ρ(Y ) U A Yt f η . With the use of these formulas identity (2.4) can be written in the following way: t=T 0= ρ(x)uΦ d x [ρ(x)u Φt − a(x)Φx u ψx (u) − b(x)h(u)Φ] d xdt − t=0 S R T dηdt = ρ(Y )U f t − ρ(Y )AU Yt f η − a(Y ) f η (ρ(Y )AU )2 U ψη (U ) ρ(Y )AU QT M f (η, t) t=T dηdt dη − − b(Y )h(U ) f t=0 A ρ(Y )AU −M QT M t=T ft f dηdt = dηdt − dη − b(Y )h(U ) f t=0 A A ρ(Y )AU QT −M QT − ρ(Y )U f η Yt + a(Y )AU ψη (U ) dηdt QT
≡ I1 − I2 + I3 + I4 .
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By the definition of A(η, t) f I1 + I3 = dηdt = I2 , A t QT which gives 0 = I4 ≡ − QT
1 Yt + a(Y )AU ψη (U ) dηdt. Φx (x, t) A x=Y (η,t)
Since Φ is arbitrary and A is separated away from zero and infinity, we conclude that the first equation of (4.6) is fulfilled a.e. in Q T . The second equation of (4.6) follows by differentiation of (4.7) with respect to η. Checking of the other properties of U and Y is straightforward. (b) Let the pair (Y, U ) be a solution of problem (4.5) in a rectangle Q T . Introduce the function U (η, t) if x = Y (η, t) with (η, t) ∈ Q T , u(x, t) = 0 otherwise. Since Y ∈ C 0 (Q T ), it is uniformly bounded in Q T . Let Φ(x, t) be an arbitrary test-function satisfying the conditions of Definition 2.1. Set f (η, t) ≡ Φ(Y (η, t), t) = Φ(x, t). Then M Y (M,t) t=T f (η, t) t=T dη = Φ(Y, t)ρ(Y )U (η, t) dY t=0 t=0 A −M Y (−M,t) t=T = Φ(x, t)ρ(x)u(x, t) d x . t=0
R
On the other hand, t M 1 d f (η, t) d b(Y ) h(U ) f (η, t) t=T dη = + f dt dηdt. exp − t=0 A dt dt QT A 0 ρ(Y ) U −M Further, QT
1 d f (η, t) dηdt = A dt
QT
= QT
1 dΦ(Y (η, t), t) dηdt A dt 1 Φt (Y, t) + Φx (Y, t)Yt dηdt A [ρ(Y )U Φt (Y, t) + Φx (Y, t)ρ(Y )U Yt ] dY dt
= QT
= QT
= ST
ρ(Y )U Φt (Y, t) − Φx (Y, t)a(Y )AUρ(Y )U ψη (U ) dY dt
ρ(x)u(x, t)Φt − Φx a(x) φx (u) d xdt
and QT
d f dt
exp − 0
t
b(Y ) h(U ) dt ρ(Y ) U
dηdt = −
f QT
b(Y ) h(U ) dηdt ρ(Y )U A
Φ(x, t)b(x)h(u) d xdt.
=− ST
Comparing these two relations, we obtain the integral identity for the function u(x, t).
5. Auxiliary propositions. The interface equation Proof of Theorem 2.1. The equivalence between problems (4.1) and (4.5) is stated in Theorem 4.1. For every fixed t ∈ [0, T ∗ (u 0 )] the function Y (η, t) is bounded and monotone increasing as a function of η, whence the existence of
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a finite limit ζ (t) ≡ Y (M, t) = lim Y (η, t). η→M−
By virtue of the first equation in (4.5) t Y (η, t) = Y (η, 0) − a(Y )U A ψη (U ) dτ 0 t a(x) ≡ Y (η, 0) − dτ, ψx (u) 0 ρ(x) x=Y (η,τ ) and the assertion follows as η → M−.
Lemma 5.1 (Monotonicity). Let condition (2.3) be fulfilled. Then the solutions of problem (4.1) are monotone decreasing in x ∈ (0, ζ (t)) for every t ∈ (0, T ∗ (u 0 )]. Proof. We argue by contradiction. Let us assume that for some t0 ∈ (0, T ∗ (u 0 )] and some points y, z ∈ (0, ζ (t0 )) (a) 0 < y < z and u(y, t0 ) = α < γ = u(z, t0 ), (b) by continuity of u(x, t), there exist the level curves lα = {(x, t) ∈ ST : u(x, t) = α}
and lγ = {(x, t) ∈ ST : u(x, t) = γ }.
Since the solution u is even, there are only two possibilities. (1) The level curve lγ connects the points (z, t0 ), (−z, t0 ) and does not touch the horizontal line t = 0. Notice that the constant function v = γ is a supersolution of Eq. (4.1). Comparing the functions u(x, t) and v = γ in the domain Dγ , bounded by the curve lγ and the line t = t0 , we arrive at the contradiction with the original hypothesis: u ≥ γ in D γ . (2) The curve lγ continues from the point (z, t0 ) to a point (s, 0). By virtue of the comparison principle the curve lγ can always be chosen in such a way that along lγ the variable t in not increasing from t0 to t = 0. Due to the symmetry of the problem there exists the level curve lγ− which connects the points (−z, t0 ), (−s, 0). Since u 0 is even and monotone decreasing in (0, l), we have that u 0 ≥ γ in [−s, s]. It follows that u ≥ γ in the domain bounded by the curves lγ , lγ− and the straight lines t = 0, t = t0 , which is impossible. Lemma 5.2 (Supersolutions). Let u be a bounded nonnegative solution of problem (4.1) in the strip ST ∗ (u 0 ) . Set µ = ∥u∥∞,ST ∗ (u ) and assume that the following conditions hold: 0 ds h(s) > 0 for s > 0, = ∞. h(s) 0+ If there exists a nonincreasing function λ(x) such that b(x) ≥ λ(x) > 0 ρ(x)
for all x > 0,
then u(x, t) ≤
Hµ−1
t
λ(ζ (τ )) dτ
in ST ,
where
0
Hµ (s) ≡
s
µ
dz . h(z)
(5.1)
Proof. Let us notice first that Hµ (s) : [0, µ] → [0, ∞] is strictly monotone decreasing and has an inverse. Denote by ω(t) the solution of the problem ω′ + λ(ζ (t))h(ω) = 0
for t > 0,
ω(0) = µ,
given by the explicit formula µ t ds Hµ (ω(t)) ≡ = λ(ζ (τ )) dτ. ω(t) h(s) 0
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We have:
1 b(x) Lω ≡ ρ(x) ω (t) − (a(x)φx (ω(t)))x + h(ω(t)) ρ(x) ρ(x) ≥ ρ(x) ω′ (t) + λ(x) h(ω(t)) ≥ ρ(x) ω′ (t) + λ(ζ (t)) h(ω(t)) = 0 ′
in the domain in D = {(x, t) : u(x, t) > 0, t ∈ (0, T ∗ (u 0 )]}, ω(t) > 0 on the boundaries {x = ζ ± (t), t ∈ [0, T ∗ (u 0 ))} of D, and ω(0) = µ ≥ u 0 in (−l, l). By the comparison principle u ≤ ω(t) in D, whence u ≤ ω(t) in ST ∗ (u 0 ) . Corollary 5.1. Let us assume that in the conditions of Lemma 5.2 b(x) ≡ λρ(x) with a positive constant λ. Then the solution of problem (4.1) satisfies the estimate u(x, t) ≤ Hµ−1 (λt)
in ST ∗ (u 0 ) .
(5.2)
The only difference in the proof is that now we take for ω(t) the solution of the problem ω′ (t) + λh(ω(t)) = 0
as t > 0,
ω(0) = µ.
Corollary 5.2. If either h ≡ 0, or b(x) ≡ 0, then estimates (5.1) and (5.2) take on the form: u≤µ
in ST ∗ (u 0 ) .
6. Upper bounds for the interface. Localized solutions 6.1. Proof of Theorem 2.2 (a) Given a solution (Y, U ) of system (4.5), let us write the first equation of (4.5) in the form ρ(Y ) Yt = −ρ(Y )AU ψη (U ), a(Y )
Y (η, 0) ∈ [0, l],
and then integrate it over the interval [0, t]: t t Y (η,t) ρ(s) ∂ ρ(Y ) Yt ds = ds dt a(s) 0 ∂t l 0 a(Y ) l t Y (η,t) ρ(s) ρ(s) = ds − ds = − ρ(Y )A U ψη (U ) dτ. a(s) l Y (η,0) a(s) 0 Choosing z = Y (η, τ ) for the new independent variable and recalling that Y (η, 0) ≤ l, we rewrite this equality as follows: z t l ρ(s) ρ(s) ρ(Y ) A U ψη (U ) η=η(z,τ ) dτ, K = ds ≤ K − ds. (6.1) a(s) 0 0 0 a(s) Integrating (6.1) over the interval (l, ζ (t)) we obtain the inequality ζ (t) z ρ(s) Jl (ζ (t)) ≡ dz ds l l a(s) ζ (t) t ≤ K (ζ (t) − l) − ρ(Y (η, τ ))A U ψη (U ) η=η(z,τ ) dτ dz l
≡ K (ζ (t) − l) −
l
0
ζ (t) t 0
ψx (u(z, τ ))|z=Y (η,τ ) dτ dz.
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Since ζ (τ ) is nondecreasing in τ and u(x, τ ) ≡ 0 for x ≥ Y (M, τ ) ≡ ζ (τ ) by definition, we may rewrite the last inequality in following form: t ζ (τ ) Jl (ζ (t)) ≤ K (ζ (t) − l) − ψx (u) dz dτ 0 l t = K (ζ (t) − l) − [ψ(u(ζ (τ ), τ )) − ψ(u(l, τ ))] dτ 0 t = K (ζ (t) − l) + ψ(u(l, τ )) dτ. 0
Hµ−1 (s) is strictly decreasing, ψ
Recall that is strictly increasing, and λ(θ ) is nonincreasing. Applying (5.1), we arrive at the estimate τ t t λ(ζ (θ )) dθ dτ ψ(u(l, τ )) dτ ≤ ψ Hµ−1 0
0
0
≤
t
ψ
0
Hµ−1 (τ λ(ζ (t)))
1 dτ = λ(ζ (t))
tλ(ζ (t))
0
ψ Hµ−1 (s) ds,
whence R(ζ (t), t) ≤ 0
for all t ≤ T ∗ (u 0 ).
Introducing the new independent variable ξ = Hµ−1 (s), we continue the last inequality as follows: µ t ψ(ξ ) 1 dξ, ψ(u(l, τ )) dτ ≤ −1 λ(ζ (t)) Hµ (tλ(ζ (t))) h(ξ ) 0 (b) If condition (2.8) is fulfilled, then the function λ(s)[Jl (s) − K (s − l)] has an inverse (for large s). Take a sequence {tk } such that tk ↑ T as k → ∞. If T < ∞, then, according to (2.5) and (2.8), |ζ (tk )| ≤ L ,
∥u(·, tk )∥ L ∞ (−L ,L) ≤ ∥u 0 ∥ L ∞ (−l,l)
uniformly with respect to k. It follows that the solution can be continued to a strip R × (T , T + ϵ], which contradicts the definition of T . (c) assume ∞Letus−1 that there exists a sequence {tk } such that ζ (tk ) → ∞ and λ(ζ (tk ))Jl (ζ (tk )) → ∞ as k → ∞. If ψ H ds < ∞, this assumption leads to the contradiction: (s) µ 0 ∞ ∞> ψ Hµ−1 (s) ds ≥ λ(ζ (tk )) [Jl (ζ (tk )) − K (ζ (tk ) − l)] → ∞, 0
which is impossible unless ζ (t) is separated away from infinity.
6.2. Proof of Theorem 2.3 If λ(x) = 0, the upper estimate on Jl (ζ (t)) has the form t Jl (ζ (t)) − K (ζ (t) − l) ≤ ψ(u(l, τ )) dτ ≤ t ψ (µ) . 0
If there exists a sequence {tk } such that Jl (ζ (tk )) → ∞ as k → ∞, then tk ≥
Jl (ζ (tk )) − K (ζ (t) − l) → ∞ as k → ∞. ψ (µ)
7. Lower bounds for the interface Lemma 7.1. Let x0 be the point chosen according to condition (2.10) and D be the constant defined in (2.11). If there exists a positive constant λ such that b(x) = λρ(x), then for every t ∈ [0, T ∗ (u 0 )] the solution of
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problem (4.5) satisfies the inequality ∞ t) d x 1 0 ρ(x)u(x, t) A(x, M ∞ U (M/2, t)A(M/2, t) ≥ ≡ 2 2D ρ(x) d x 0
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(7.1)
with the constant M from condition (2.3). Proof. We argue by contradiction. Let us assume that for some t0 ∈ [0, T ∗ (u 0 )] and all η ∈ [M/2, M] U (η, t0 , ) A(η, t0 ) < M/2K . According to the second equation of (4.5) (the mass balance law) ∞ ζ (t0 ) Y (M,t0 ) D= ρ(s) ds > ρ(x) d x = ρ(s) ds 0
0 M
= 0
ds ≥ A(s, t0 )U (s, t0 )
0
M M/2
ds M 2D > = D, A(s, t0 )U (s, t0 ) 2 M
which contradicts the assumption. Let us fix an arbitrary t ∈ [0, T ∗ (u 0 )] and choose a point η′ ∈ [M/2, M] such that U (η′ , t)A(η′ , t) ≥ M/2D. Recall that h(U )/U is an increasing function of U , U (η, t) and Y (η, t) are decreasing functions of η, and λ(s) is a decreasing function. Then t t h(U ) h(U ) dτ ≤ exp λ(Y ) dτ = A(M/2, t), A(η′ , t) = exp λ(Y ) U (η′ ,τ ) U (M/2,τ ) 0 0 whence U (M/2, t)A(M/2, t) ≥ U (η′ , t)A(η′ , t) ≥ M/2D. Corollary 7.1. If either h ≡ 0, or b(x) ≡ 0, then estimate (7.1) reads as follows: U (M/2, t) ≥ M/2D for all t ∈ [0, T ∗ (u 0 )]. 7.1. Proof of Theorem 2.4 (a) Let us write the first equation of (4.5) in the form Yt = −ψη (U ) a(Y ) U A and then integrate it over the interval [0, t]: Y (η,t) Y (η,0) t ds ds = − ψη (U ) dτ, a(Y ) U A a(Y ) U A x0 x0 0 where x0 is the point chosen in (2.10). Integration of this equality in the limits [M/2, M] gives M Y (η,t) M Y (η,0) M t ds ds dη − ψη (U ) dτ dη. = a(Y ) U A a(s) u A M/2 x0 M/2 x0 M/2 0 Dropping the nonnegative terms on the right-hand side, we get M Y (η,t) t ds dη ≥ ψ(U (M/2, t)) dτ. a(s) u A M/2 x0 0 t) is monotone decreasing in x, we conclude that Since u A(x, M Y (η,t) t 1 ds I ≡ dη ≥ ψ(U (M/2, τ )) dτ. (η, t), t) x0 a(s) M/2 u(Y (η, t), t) A(Y 0
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Reverting to the variables (x, t), we find that G x0 (ζ (t)) = I . By Lemmas 5.2 and 7.1 the following estimate holds: τ h(U ) M dθ ψ (U (M/2, τ )) ≥ ψ exp − λ(Y ) 2D U (M/2,θ) 0 τ M h(ω) ≥ψ exp − (θ ) dθ λ(Y ) 2D ω 0 τ ′ ω M exp (θ ) dθ =ψ 2D ω 0 M ω(τ ) M =ψ =ψ exp ln ω(τ ) . 2D ω(0) 2µD It follows that G x0 (ζ (t)) ≥
0
t
ψ
M H −1 (λτ ) dτ 2µD µ
=
1 λ
0
λt
ψ
M H −1 (s) ds 2µD µ
= F(t).
(b) Let G x0 (∞) = q < ∞. By the assumption there exists a sequence {tk } such that tk ↗ ∞, ζ (tk ) < ∞ and F(tk ) → ∞ as k → ∞. Along this sequence q ≥ G x0 (ζ (tk )) ≥ F(tk ) → ∞ as k → ∞, which is impossible unless there exists T such that ζ (t) → ∞ as t ↗ T . (c) In this case we apply Corollary 7.1: t M M dτ = t ψ . G x0 (ζ (t)) ≥ ψ 2D 2D 0 (d) In case that G x0 (∞) < ∞, the above inequality is possible only if ζ (t) → ∞ as t ↗ T ∗ .
Acknowledgment First author acknowledges the support of the Research Grant MTM2010-18427, MICINN, Spain References [1] U.G. Abdulla, Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption, Nonlinear Anal. 50 (4) (2002) 541–560. URL: http://dx.doi.org/10.1016/S0362-546X(01)00764-7. [2] U.G. Abdulla, J.R. King, Interface development and local solutions to reaction–diffusion equations, SIAM J. Math. Anal. 32 (2) (2000) 235–260. [3] S.N. Antontsev, J.I. D´ıaz, S. Shmarev, Energy Methods for Free Boundary Problems, in: Progress in Nonlinear Differential Equations and their Applications, vol. 48, Birkh¨auser, Boston, Inc., Boston, MA, 2002, URL: http://dx.doi.org/10.1007/978-1-4612-0091-8. [4] S.N. Antontsev, S.I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal. 60 (3) (2005) 515–545. URL: http://dx.doi.org/10.1016/j.na.2004.09.026. [5] S.N. Antontsev, S.I. Shmarev, Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity, Fundam. Prikl. Mat. 12 (4) (2006) 3–19. URL: http://dx.doi.org/10.1007/s10958-008-0129-6. [6] J.I. D´ıaz, S. Shmarev, Lagrangian approach to the study of level sets: application to a free boundary problem in climatology, Arch. Ration. Mech. Anal. 194 (1) (2009) 75–103. URL: http://dx.doi.org/10.1007/s00205-008-0164-y. [7] D. Eidus, The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium, J. Differential Equations 84 (2) (1990) 309–318. URL: http://dx.doi.org/10.1016/0022-0396(90)90081-Y. [8] D. Eidus, S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc. 120 (3) (1994) 825–830. URL: http://dx.doi.org/10.2307/2160476. [9] B.H. Gilding, R. Kersner, The characterization of reaction-convection–diffusion processes by travelling waves, J. Differential Equations 124 (1) (1996) 27–79. URL: http://dx.doi.org/10.1006/jdeq.1996.0002. [10] B.H. Gilding, R. Kersner, Travelling waves in nonlinear diffusion–convection reaction, in: Progress in Nonlinear Differential Equations and their Applications, vol. 60, Birkh¨auser, Verlag, Basel, 2004, URL: http://dx.doi.org/10.1007/978-3-0348-7964-4. [11] G. Grillo, M. Muratori, M.M. Porzio, Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincar´e inequalities, Discrete Contin. Dyn. Syst. 33 (8) (2013) 3599–3640. URL: http://dx.doi.org/10.3934/dcds.2013.33.3599. [12] M. Guedda, D. Hilhorst, M.A. Peletier, Disappearing interfaces in nonlinear diffusion, Adv. Math. Sci. Appl. 7 (2) (1997) 695–710.
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[13] M. Guedda, D. Hilhorst, M.A. Peletier, Blow-up of interfaces for an inhomogeneous aquifer, in: Free Boundary Problems: Theory and Applications, (Crete, 1997), in: Res. Notes Math., vol. 409, Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 131–138. [14] M. Guedda, D. Hilhorst, S.I. Shmarev, Regularity of interfaces for an inhomogeneous filtration equation, Adv. Nonlinear Stud. 1 (1) (2001) 80–101. [15] R.G. Iagar, G. Reyes, A. S´anchez, Radial equivalence of nonhomogeneous nonlinear diffusion equations, Acta Appl. Math. 123 (2013) 53–72. URL: http://dx.doi.org/10.1007/s10440-012-9714-2. [16] R.G. Iagar, A. S´anchez, Asymptotic behavior for the heat equation in nonhomogeneous media with critical density, Nonlinear Anal. 89 (2013) 24–35. URL: http://dx.doi.org/10.1016/j.na.2013.05.002. [17] A.S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk 42 (2(254)) (1987) 135–176, 287. [18] S. Kamin, R. Kersner, Disappearance of interfaces in finite time, Meccanica 28 (2) (1993) 117–120. URL: http://dx.doi.org/10.1007/ BF01020323. [19] S. Kamin, R. Kersner, A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 9 (4) (1998) 279–298. 1999. [20] S. Kamin, G. Reyes, J.L. V´azquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density, Discrete Contin. Dyn. Syst. 26 (2) (2010) 521–549. URL: http://dx.doi.org/10.3934/dcds.2010.26.521. [21] S. Kamin, P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math. 34 (6) (1981) 831–852. URL: http://dx.doi.org/10.1002/cpa.3160340605. [22] S. Kamin, P. Rosenau, Nonlinear thermal evolution in an inhomogeneous medium, J. Math. Phys. 23 (7) (1982) 1385–1390. URL: http://dx.doi.org/10.1063/1.525506. [23] R. Kersner, G. Reyes, A. Tesei, On a class of parabolic equations with variable density and absorption, Adv. Differential Equations 7 (2) (2002) 155–176. [24] A.M. Meirmanov, V.V. Pukhnachov, S.I. Shmarev, Evolution Equations and Lagrangian Coordinates, in: de Gruyter Expositions in Mathematics, vol. 24, Walter de Gruyter & Co., Berlin, 1997, URL: http://dx.doi.org/10.1515/9783110874440. [25] S. Nieto, G. Reyes, Asymptotic behavior of the solutions of the inhomogeneous porous medium equation with critical vanishing density, Commun. Pure Appl. Anal. 12 (2) (2013) 1123–1139. URL: http://dx.doi.org/10.3934/cpaa.2013.12.1123. [26] L.A. Peletier, A. Tesei, Diffusion in inhomogeneous media: localization and positivity, Ann. Mat. Pura Appl. 141 (4) (1985) 307–330. URL: http://dx.doi.org/10.1007/BF01763179. [27] G. Reyes, A. S´anchez, Disappearance of interfaces for the porous medium equation with variable density and absorption, Asymptot. Anal. 36 (1) (2003) 13–20. [28] G. Reyes, J.L. V´azquez, The Cauchy problem for the inhomogeneous porous medium equation, Netw. Heterog. Media 1 (2) (2006) 337–351. URL: http://dx.doi.org/10.3934/nhm.2006.1.337. [29] G. Reyes, J.L. V´azquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal. 8 (2) (2009) 493–508. URL: http://dx.doi.org/10.3934/cpaa.2009.8.493. [30] P. Rosenau, S. Kamin, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math. 35 (1) (1982) 113–127. URL: http://dx.doi.org/ 10.1002/cpa.3160350106. [31] S. Shmarev, Interfaces in solutions of diffusion–absorption equations in arbitrary space dimension, in: Trends in Partial Differential Equations of Mathematical Physics, in: Progr. Nonlinear Differential Equations Appl., vol. 61, Birkh¨auser, Basel, 2005, pp. 257–273. URL: http://dx.doi.org/10.1007/3-7643-7317-2 19. [32] S.I. Shmarev, J.L. V´azquez, Lagrangian coordinates and regularity of interfaces in reaction–diffusion equations, C. R. Acad. Sci. Paris I Math. 321 (8) (1995) 993–998. [33] S.I. Shmarev, J.L. Vazquez, The regularity of solutions of reaction–diffusion equations via Lagrangian coordinates, NoDEA Nonlinear Differential Equations Appl. 3 (4) (1996) 465–497. URL: http://dx.doi.org/10.1007/BF01193831. [34] A.F. Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations, Appl. Anal. 86 (6) (2007) 755–782. URL: http://dx.doi.org/10.1080/00036810701435711. [35] J.L. V´azquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
ScienceDirect Mathematics and Computers in Simulation (
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Original articles
Interfaces in diffusion–absorption processes in nonhomogeneous media Sergey Shmarev a,∗ , Viktor Vdovin b , Alexey Vlasov b a Mathematics Department, University of Oviedo, Oviedo, Spain b Mathematics Department, Siberian Transport University, Novosibirsk, Russia
Received 19 March 2014; received in revised form 18 September 2014; accepted 20 November 2014
Abstract We study the Cauchy problem for the nonlinear parabolic equation
ρ(x)u t = (a(x)φx (u))x − b(x)h(u)
in R × (0, T ]
with nonnegative coefficients ρ(x), a(x) and b(x). It is assumed that φ(0) = 0, φ ′ (s) > 0, φ ′ (s)/s ∈ L 1 (0, δ) for some δ > 0, h(s) ≥ 0 and h(s)/s is nondecreasing for s ≥ 0. The solution of this problem may possess the property of finite speed of propagation of disturbances from the data, which leads to formation of interfaces that bound the support of the solution. It is proved that the behavior of interfaces can be characterized in terms of convergence or divergence of the integrals x
ρ(s)
x0
s x0
dz a(z)
ds,
Jx0 (x) =
b(x) ρ(x)
x x0
0
s
ρ(z) dz a(z)
ds,
b(x) Jx (x), ρ(x) 0
x
ρ(s) ds
x0
as x → ∞ and
ϵ
ds , h(s)
ϵ
ψ(s) ds h(s)
as ϵ → 0 + .
We derive two-sided a priori bounds for the interface location, establish sufficient and necessary conditions for disappearance of interfaces in a finite time (the interface blow-up), and derive the integral equation for the interface. c 2014 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights ⃝ reserved. Keywords: Inhomogeneous diffusion–absorption equation; Blow-up of interfaces; Localized solutions; Lagrangian coordinates
1. Introduction We study the behavior of nonnegative weak solutions of the degenerate parabolic equation ρ(x)u t = (a(x)φx (u))x − b(x)h(u).
(1.1)
∗ Corresponding author.
E-mail addresses: [email protected] (S. Shmarev), [email protected] (V. Vdovin), [email protected] (A. Vlasov). http://dx.doi.org/10.1016/j.matcom.2014.11.004 c 2014 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights 0378-4754/⃝ reserved.
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The coefficients ρ(x), a(x), b(x) are given functions, a(x) and ρ(x) are strictly positive on every compact subset of R \ {0}, b(x) is nonnegative in R. The functions φ(s), h(s) : R+ → R+ are assumed to satisfy the conditions φ ′ (s) φ(0) = 0, φ ′ (s) > 0 for a.a. s > 0, ds < ∞, s 0+ h(s) ≥ 0,
h(s) is nondecreasing s
for s ≥ 0.
(1.2)
(1.3) (1.4)
Eq. (1.1) appears in the mathematical description of various physical phenomena such as the heat propagation in inhomogeneous medium [21,30], thermal evolution in heated plasma [22], joint motion of salt and fresh water [12,13]. This equation generalizes the porous medium equation u t = (u m )x x ,
m > 1.
(1.5)
Unlike the linear heat equation, the solutions of (1.5) possess the property of finite speed of propagation of disturbances from the data, which leads to appearance of the interfaces. The interfaces are plain curves x = ζ (t) ≡ sup{x ∈ R : u(x, t) > 0},
x = ξ(t) ≡ inf{x ∈ R : u(x, t) > 0},
that separate the regions where the solution is positive from those where it equals zero. It is known that in the solutions of (1.5) the interfaces propagate with finite speed, never stop, and tend to infinity as t → ∞, see, e.g., [17,35]. This type of behavior is typical for the solutions of Eq. (1.1) with constant positive coefficients ρ, a, b, which can be regarded as a model of the process of slow diffusion (condition (1.3)) with weak absorption (condition (1.4)). The situation changes if we consider the inhomogeneous counterpart of Eq. (1.1). Let us consider the Cauchy problem for the equation ρ(x)u t = (u m )x x in R × (0, T ), (1.6) u(x, 0) = u 0 (x) in R, supp u 0 = (−l, l), 0 < l < ∞, assuming that the coefficient ρ(x) is strictly positive on every compact subset of R. It was discovered in [18] that problem (1.6) admits solutions whose interfaces vanish in a finite time, that is, the initially bounded set supp u(x, ·) becomes infinite at a finite moment t = T ∗ . A characterization of this effect in terms of the properties of ρ(x) is given in [21,30]. It is shown that if ρ(x) and u 0 (x) are even functions, then the following alternative holds: ∞ <∞ ⇒ supp u(x, t) = R at a finite instant T ∗ (the so-called interface blow-up), xρ(x)d x =∞ ⇒ supp u(x, t) remains finite for all t > 0. 1 Less is known about the phenomenon of the interface blow-up in the multidimensional case. It is proved in [12] that for the solutions of the Cauchy problem for the equation ρ(|x|)u t = ∆u m ,
m > 1,
(1.7)
in R2 × (0, T ] the set supp u(x, t) becomes unbounded in a finite time if ∞ s ρ(s) ln s ds < ∞ 1
and remains bounded for all t > 0 if ∞ ln s min ξ 2 ρ(ξ ) ds = ∞. s ξ ∈(0,s) 1 In the case n ≥ 3 the effect of the interface disappearance in solutions of (1.7) is studied in [34]. It is shown that the interfaces in solutions of Eq. (1.7) with ρ(x) ∼ x −λ as x → ∞ disappear in a finite time if λ > n(m−1)+2 , and remain m
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bounded for all finite t > 0 if λ < n(m−1)+2 . Paper [34] deals with the doubly nonlinear equations of the form m ρ(|x|)u t = div |u|m−1 |∇u|λ−2 ∇u , u t = div a(|x|)|u|m−1 |∇u|λ−2 ∇u , which contain (1.7) as a partial case. The analysis is based on the study of the local energy estimates in weighted Sobolev spaces. The behavior of interfaces and the possibility of the interface blow-up in solutions of the diffusion–absorption equation ρ(x)u t = (u m )x x − cu p with m > 1, p > 0 and c = const > 0 is studied in papers [23,27]. It is shown that the interfaces remain bounded p−1) as for all t > 0 if either p ∈ (0, m], or p > m and ρ(x) ∼ |x|−k as x → ∞ with k ≤ k ∗ = 2(p−m . If p > m and k > k ∗ , then the interfaces disappear in a finite time. The behavior of interfaces in solutions of the Cauchy problem for the diffusion–absorption equation with variable absorption, u t = ∆u m − b(x)u p
in Rn × (0, ∞],
is studied in paper [26]. It is shown that for p ≥ m the support of the solution covers the whole of Rn as t → ∞ and, conversely, is contained in a finite ball for all t > 0 (localization of the solution) if p ∈ [1, m) and b(x) ∼ |x|−k as |x| → ∞ with k ≥ 2. For k > 2 and p ∈ [1, m) the solution is localized if the initial datum is appropriately small. The aim of present work is to describe the behavior of interfaces in solutions of Eq. (1.1) in terms of convergence and divergence of the integrals s x s x dz b(x) b(x) x ρ(z) G x0 (x) = ρ(s) ds , Jx0 (x) = dz Jx (x), ds ρ(s) ds (1.8) ρ(x) x0 ρ(x) 0 x0 x0 a(z) 0 a(z) x0 as x → ∞ and the behavior of the integrals µ s ′ µ 1 φ (z) ds , dz ds h(s) h(s) z ϵ 0 ϵ
as ϵ → 0 + .
(1.9)
The parameter µ > 0 is defined in condition (2.3). We prove that the behavior of interfaces in solutions of the inhomogeneous equation (1.1) with finitely supported initial data depends only on the nonlinear structure of the equation, which is described in terms of convergence of the integrals in (1.8), (1.9). By now there exist exhaustive studies of the property of finite speed of propagation from the data and the behavior of supports in solutions of homogeneous equations of the type diffusion–absorption with constant a and ρ and b- see, e.g., [9,10] for pertinent results and a review of the literature. We refer also to [1,2], for an analysis of the short-time behavior of interfaces in solutions of equations with power nonlinearities φ(s) = s m and h(s) = s β with m, β > 0. Notice also that since the radially symmetrical solutions of the multidimensional equation ρ(|x|)u t = div (a(|x|)∇φ(u)) − b(|x|)h(u)
in Rn × (0, T ]
(1.10)
satisfy a one-dimensional equation of the type (1.1), the results of the present work may be applied, by comparison, to solutions of certain equations of the type (1.10). 2. Assumptions and results Let us consider the Cauchy problem ρ(x)u t = (a(x) u ψx (u))x − b(x)h(u) u(x, 0) = u 0 (x) in R
in ST = R × (0, T ],
(2.1)
with the initial function u 0 (x) > 0
for x ∈ (−l, l), l > 0,
u 0 (x) = 0
in R \ (−l, l)
(2.2)
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and the coefficients ρ(x), a(x) and b(x) ≥ 0 satisfying (1.2). The function ψ(·) is defined by the relation s ′ φ (z) ψ(s) = dz, z 0 so that s ψ ′ (s) = φ ′ (s), ψ(0) = 0, and Eq. (2.1) is equivalent to (1.1). Throughout the paper we assume that the data of problem (2.1) satisfy (1.2) and the following conditions: ρ, a, b, u 0 are even continuous functions in R, is nonincreasing for x > 0, u0 ∞ ρ(x)u 0 (x) d x = M < ∞, 0 µ = sup u < ∞.
(2.3)
0
R
Definition 2.1. A function u(x, t) is called weak solution of problem (2.1) in ST if 1. u is nonnegative, continuous and uniformly bounded in ST , ρu ∈ L ∞ (R) for all t ∈ [0, T ], 2. √a u ψx (u) ∈ L 2 (S ), T 3. for every smooth test-function Φ, vanishing for all sufficiently large |x| t=T = 0. I (u, Φ) ≡ ρuΦ d x [ρu Φt − aΦx u ψx (u) − Φbh(u)] d xdt − ST
R
(2.4)
t=0
If in the conditions of Definition 2.1 I (u, Φ) ≤ 0 (I (u, Φ) ≥ 0) for every nonnegative test-function Φ, then the function u is called supersolution (subsolution) of problem (2.1). Under suitable assumptions on the data, existence of solutions to problem (2.1) can be proved by known methods— see, e.g., [8,12,19,23,28]. The cited papers use different notions of weak solution, but in every case the solution is constructed as the limit of a sequence of solutions to the problems posed in expanding cylinders with zero Dirichlet conditions on the lateral boundary. Such a procedure leads to the minimal solution of the Cauchy problem. It is noteworthy that in the case when ρ(x) is rapidly increasing as x → ∞, global in time solutions to problem (2.1) may be not unique. We refer to [7,8,19] for more information on this issue. To formulate the a priori results about the interface behavior we assume that the Cauchy problem (2.1) admits a local in time solution which possesses the property of finite speed of propagation of disturbances from the initial data. Namely, we claim that the following assertion holds. Let conditions (1.2)–(1.4), (2.2), (2.3) be fulfilled. Then there exists T ∗ (u 0 ) > 0 such that problem (2.1) has in the strip ST ∗ (u 0 ) a solution u(x, t) that possesses the properties: ∀ τ ∈ (0, T ∗ (u 0 )] ∥u(·, τ )∥∞,R ≤ µ = sup u 0 ,
(2.5)
R
ζ (t) = sup{x ∈ R : u(x, t) > 0} ≤ L
on [0, T ∗ (u 0 )], L = const > 0.
All further considerations will be confined to the time interval [0, T ] with T = sup{T ∗ (u 0 ) > 0 : (2.5) is fulfilled}. Properties (2.5) are typical for solutions of parabolic equations which display the finite speed of propagation and obey the maximum principle. For the sake of completeness of presentation, in Section 3.1 we outline a proof of (2.5) for subclasses of Eq. (1.10). The main results of this work are the following. Theorem 2.1 (The Interface Equation). Let conditions (1.2)–(1.4), (2.2), (2.3) and (2.5) be fulfilled. Then for every t ∈ [0, T ∗ (u 0 )) there exists the limit t a(x) lim ψx (u(x, τ )) dτ x→ζ (t) 0 ρ(x)
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and the following interface equations hold: t a(x) ζ (t) = ζ (0) − lim ψx (u(x, τ )) dτ. x→ζ (t) 0 ρ(x)
5
(2.6)
Since the solution of problem (2.1) is nonnegative, it follows from the interface equation (2.6) that ζ (t) is a monotone increasing function. Accept the notations l µ ρ(x) ds K = d x, Hµ (z) ≡ (2.7) a(x) h(s) 0 z with the constants l > 0 and µ > 0 from conditions (2.2) and (2.3). Theorem 2.2 (Localized Solutions). Let the conditions of Theorem 2.1 be fulfilled. Assume that h(s) > 0 for s > 0 and for all x ≥ 0, b(x) ≥ λ(x) > 0 with a nonincreasing function λ(x) ρ(x) and introduce the function R(σ, t) ≡ λ(σ ) [Jl (σ ) − K (ζ (t) − l)] −
tλ(σ )
0
ψ Hµ−1 (s) ds
with Jl (x) defined in (1.8) and the constant K from (2.7). The following assertions hold. (a) Problem (2.1)–(2.2) has a solution satisfying the inequality ∀ t ∈ [0, T )
R(ζ (t), t) ≤ 0.
(b) If the function λ(s) [Jl (s) − K (s − l)] is monotone increasing as s → ∞ and ∀t >0
sup {s > 0|R(s, t) < 0} < ∞,
(2.8)
then T = ∞, i.e., ζ (t) → ∞ only as t → ∞. (c) If lim sup λ(s) [Jl (s) − K (s − l)] = ∞ and s→∞
0
∞
ψ Hµ−1 (s) ds < ∞,
then T = ∞ and the solution is localized, i.e., there exists a finite constant L > 0 such that ζ (t) ≤ L for all t > 0. Theorem 2.3 (Finite Speed of Propagation). Let the conditions of Theorem 2.1 be fulfilled. If b(x)h(u) ≡ 0, then problem (2.1)–(2.2) admits a solution such that ζ (t) → ∞ only as t → ∞, i.e., T = ∞, and t ∀ t > 0 Jl (ζ (t)) ≤ K (ζ (t) − l) + ψ (u(0, τ )) dτ (2.9) 0
with the constants l and K defined in (2.2) and (2.7). The next assertion establishes lower bounds for the interface location and gives sufficient conditions for the interface blow-up. Let us fix a point x0 ∈ (0, l) ≡ supp u 0 ∩ {x > 0} such that x0 M ρ(x)u 0 (x) d x = (2.10) 2 0 with the constant M from (2.3). For the initial data u 0 satisfying (2.3) such a choice is always possible. Theorem 2.4 (Blow-up of Interfaces). Let the conditions of Theorem 2.1 be fulfilled, and, additionally, ∞ ρ(x) d x = D < ∞. 0
(2.11)
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(a) If h(s) > 0 for s > 0 and b(x) = λρ(x) with a positive constant λ, then problem (2.1)–(2.2) has a solution u(x, t) such that ζ (t) s dz ∗ ∀ t ∈ [0, T (u 0 )) G x0 (ζ (t)) ≡ ρ(s) ds x0 x0 a(z) λt 1 M −1 ≥ F(t) ≡ ψ (2.12) H (s) ds. λ 0 2D µ (b) If G x0 (∞) < ∞ and lim supt→∞ F(t) = ∞, then the interface disappears in a finite time, i.e., ζ (t) → ∞ as t ↗ T. (c) If b(x)h(s) ≡ 0 and G x0 (∞) = ∞, then the lower bound for the interface is given by the inequality M . ∀ t ∈ [0, T ] G x0 (ζ (t)) ≥ t ψ 2D (d) If b(x)h(s) ≡ 0 and G x0 (∞) < ∞, then the interface disappears at an instant T ≤
G x0 (∞) . ψ(M/2D)
The proofs of Theorems 2.1–2.4 are based on the nonlocal coordinate transformation that renders the interface stationary. This change of independent variables, frequently used in continuum mechanics, is explained in Section 4. 3. Examples Let us illustrate the assertions of Theorems 2.1–2.4 by the examples of model equations of the type (1.1). In these examples the integrals G x0 (x) and Jl (x) can be calculated explicitly, which provides a possibility to compare the above results with some of the already known results obtained with other methods. 3.1. Local in time solvability Let us consider the Cauchy problem for the equation ρ(x)u t = ∆ u m − b(x)u p
in ST = Rn × (0, T ], m > 1, p ≥ 1,
(3.1)
with a nonnegative initial function u 0 such that supp u 0 ⊂ B1 (0). Assume that the coefficients a(x) and ρ(x) are strictly positive on every compact subset of Rn , but may vanish as |x| → ∞. Eq. (3.1) is of the type (1.10), or (1.1) in the special case of radially symmetric coefficients ρ and b. A solution to this problem can be constructed in a number of ways. For example, it can be done by adapting the arguments of [4,5]. Fix some r > 1 and consider the Dirichlet problem vt = ∆ A(x)v m − B(x)v p in Q r,T = Br (0) × (0, T ], v = 0 on the lateral boundary of Q r,T , (3.2) v(x, 0) = v0 ≡ ρ(x)u 0 (x) in Br (0). Here T > 0 is to be defined and the coefficients have the form A(x) = ρ −m (x),
B(x) = b(x)ρ − p (x).
Problem (3.2) follows from (3.1) after the substitution v = ρ u. This problem is a partial case of the problem considered in [4,5]. Since ρ is strictly positive in Br (0), then A(x), B(x) ∈ C 0 (Br (0)), A(x) is separated away from zero and infinity in Br (0), and B(x) ≥ 0 is bounded. By [4,5], for every nonnegative initial function u 0 ∈ L ∞ (Br (0)) and every T > 0 problem (3.2) has in the cylinder Q r,T a unique nonnegative solution v(x, t) such that ∥v∥ L ∞ (Qr,T ) ≤ ∥v0 ∥ L ∞ (Br (0)) ≡ ∥ρ u 0 ∥ L ∞ (Br (0)) ,
∥∇v m ∥22,Qr,T ≤
1 ∥v0 ∥22,Br (0) . 2
(3.3)
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The solutions of problem (3.2) possess the property of finite speed of propagation of disturbances from the data—see [4], [3, Ch. 3, Th. 2.1]: if v0 (x) ≡ 0 in a ball Bs0 (x0 ) ⊂ Br (0), then v(x, t) ≡ 0 in the co-centric smaller ball Bs(t) (x0 ) of radius 1 1 s(t) = s0ν − C t γ ν for t ≤ s0ν /C ν . The positive parameters ν, γ , C depend on m, p, the norm ∥v0 ∥(m+1)/m,Bs0 (x0 ) and the quotient sup ρ(x)/ inf ρ(x).
Bs0 (x0 )
Bs0 (x0 )
In particular, C ↗ ∞ as inf{ρ(x)|x ∈ Bs0 (0)} ↘ 0. It follows that for every big r > 0 and the sufficiently small T the support of the solution does not reach the lateral boundary of the cylinder Q r,T . Continuing the solution by zero from Q r,T to the rest of the strip ST , we obtain a solution of the Cauchy problem. The first estimate in (3.3) allows us to continue the solution to the strip Rn × (T, T1 ]. This process is continued until the moment when either the support Ti = ∞. In case that the initial function u 0 and the coefficients ρ, a, b of the solution becomes unbounded, or are radially symmetric, the constructed solution of Eq. (3.1) is also radially symmetric and satisfies the conditions of Definition 2.1 in the strip ST . For the equation u t = ∆φ(u) with φ(s) satisfying conditions (1.3), existence of radially symmetric solutions satisfying (2.5) is proved in [24, Ch. 3, Sec. 25]. We refer also to the papers [16,15,25,20,29] for a discussion of solvability of the equation ρ(x)u t = ∆φ(u). The weighted porous medium equation ρν u t = div ρµ ∇u m (3.4) with m > 1 and positive functions ρν , ρµ was considered in [11]. 3.2. The porous medium equation Let us consider the Cauchy problem for the porous medium equation u t = ∆u m
in Rn × R+ , m > 1.
(3.5)
This equation admits the class of explicit self-similar solutions (the Barenblatt solutions) 1/(m−1) x U (x, t) = (1 + t)−α A − B|ξ |2 , ξ= , + (1 + t)β
(3.6)
with the parameters α=
1 , m − 1 + 2/n
β=
α , n
B=
m−1 β, m
A = const.
For these solutions U m−1 (0, t) = A(1 + t)−α(m−1) . The function U is a solution of the radially symmetrical version of Eq. (3.5) r n−1 Ut = r n−1 (U m )r , r = |x| > 0, r
which coincides with (2.1) if we choose ρ(r ) ≡ a(r ) = r n−1 , h ≡ 0. By Theorem 2.3, the interface of the self-similar solution obeys the estimate 2 t m 1 ζ (t) − A/B ≤ A/B ζ (t) − A/B + U m−1 (0, τ ) dτ. 2 0 m−1 It follows that 2 ζ (t) − 2 A/B ≤ A/B + 2 0
t
m U m−1 (0, τ ) dτ ∼ (1 + t)2β m−1
as t → ∞.
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This example shows that estimate (2.9) is sharp because the interface of the self-similar solution (3.6) is the plane curve defined by the explicit formula r = ζ (t) ≡ A/B(1 + t)β , r = |x|. 3.3. The porous medium equation with variable density In the case of the Cauchy problem for the equation ρ(x)u t = ∆ φ(u)
in Rn × R+
(3.7)
with φ(·) and ρ(·) satisfying conditions (1.2) and (1.3) Theorem 2.4 gives the following conditions for the interface blow-up: ∞ xρ(x) d x < ∞ if n = 1, 1 ∞ the interfaces disappear x ln x ρ(x) d x < ∞ if n = 2, ⇒ in a finite time 1 ∞ x n−1 ρ(x) d x < ∞ if n ≥ 3 1
(compare with [12,21,30]). Under various conditions on the density ρ(x), the long time behavior of solutions of Eq. (3.7) was studied in [16,15,25,20,29]. 3.4. Localized solutions of the diffusion–absorption equation Let us consider the diffusion–absorption equation with power nonlinearities ρ(|x|)u t = ∆u m − b(|x|)u p
in Rn × R+ ,
(3.8)
assuming that m > 1, p > 0, ρ(|x|) = (1 + |x|)−α
and
b(|x|) = (1 + |x|)−β ,
α, β ≥ 0.
(3.9)
For Eq. (3.8) under assumptions (3.9) b(r ) ≥ λ(r ) ≡ (1 + r )α−γ ρ(r )
with γ = max{β, α}.
Let us assume, for the sake of definiteness, that the support of the initial function is contained in the ball B1 (0). A straightforward calculation gives: for 0 < l < 1 and µ = supR u 0 x α−γ +1 if α > 1, 1+x 2−γ as x → ∞ λ(x)[Jl (x) − K (x − l)] ≥ C x ln if α = 1, 1+l 2−γ x if α ∈ [0, 1), with C ≡ C(α, γ ) and s
0
p−m s p−1 if p > m, ψ Hµ−1 (−s) ds ∼ ln(1 + s) if p = m, const if p < m.
It follows that the solutions of (3.8) are localized if either α ∈ [0, 1] and β < 2, p < m and or α > 1 and β < α + 1 (compare with [23,26,27]).
(3.10)
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3.5. Disappearance of interfaces in diffusion–absorption equations Let us notice first that since the solutions of Eq. (3.7) with φ(s) = s m , m > 1, are supersolutions of Eq. (3.8), the inequality G x0 (∞) < ∞ is necessary for the blow-up of interfaces in solutions of (3.8). Let condition (3.9) be fulfilled and b(x) = λρ(x). By Theorem 2.4 the support of the solution becomes unbounded in a finite time if ∞ s dz < ∞, p ≥ m and G x0 (∞) = s n−1 (1 + s)−α ds n−1 x0 x0 z that is, if α >
2 n
for n = 1, 2, for n ≥ 3.
Condition (2.11) is fulfilled if in condition (3.9) α > 1. Comparison of solutions of the equations ρ(|x|)u t = ∆u m − λρ(|x|)u p
and
ρ(|x|)u t = ∆u m − b(x)u p
shows that in the solutions of the latter the interfaces disappear in a finite time if 2 for n = 1, 2, p ≥ m and β ≥ α > n for n ≥ 3. 3.6. Eventual positivity Let in Eq. (3.8) p ≥ m, b(x) = λρ(|x|) with some λ = const > 0 and let b, ρ are of the form (3.9). Let us assume, for the sake of definiteness, that n ≥ 3, G x0 (∞) = ∞, and claim that condition (2.11) is fulfilled: 1<α ≤ n
if n ≥ 3.
Under these assumptions the interfaces in solutions of (3.8) cannot display the blow-up behavior and estimate (2.12) holds. Moreover, by virtue of (2.6) the function ζ (t) is monotone nondecreasing. Plugging (3.10) into (2.12), we estimate ζ (t) from below: p−m ζ n−α (t) − x0n−α if n > α, p−1 t if p > m, 1 + ζ (t) C0 ≤ G x0 (ζ (t)) ≤ C1 if n = α ln ln(1 + t) if p = m 1 + x0 with finite constants C0 = C0 ( p, m) and C1 = C1 (α, n, x0 ). 4. The change of independent variables. Lagrangian formulation In this section we introduce the change of independent variables which allows us to reduce free boundary problems for parabolic equations to equivalent problems posed in a rectangular time-independent domain. Let us interpret the Cauchy problem ρ(x)u t = (a(x)φx (u))x − b(x)h(u) in ST , (4.1) u(x, 0) = u 0 (x) in R, supp u 0 = (−l, l), l < ∞, as the mathematical description of the one-dimensional motion of a fluid with the density d(x, t) = ρ(x)u(x, t)
and the velocity v = −
a(x) ψx (u). ρ(x)
Eq. (4.1) expresses the mass balance law in the Euler description, dt + (d v)x = −b(x)h(u), where the term −b(x)h(u) represents the density of the mass forces. The initial condition in (4.1) defines the initial distribution of density: d(x, 0) = ρ(x)u 0 (x) in R.
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There are two methods of description of motions of continua. In the Euler method the characteristics of motion are considered as functions of the time t and a coordinate system not connected with the medium itself. In the method of Lagrange the main quantities characterizing the motion are considered as functions of the time t and the initial positions of the particles. The latter method is of special convenience if the domain occupied by the moving medium is a priori unknown and has to be defined together with the other characteristics of motion. If the medium is constituted by the same particles at each instant, on the plane of Lagrangian coordinates the flow domain is time-independent. From the mathematical point of view, the existence of these two counterpart descriptions of the same mechanical process furnishes a nonlocal coordinate transformation that renders the free boundary stationary (see [24] for more information on this issue). This change of independent variables was already used to study the interface regularity in solutions of problem (1.6) [14]. In the present work we follow the method of introduction of Lagrangian coordinate proposed in [32,33] for the study of interfaces in processes without mass conservation. A local version of the system of Lagrangian coordinates was used in [31,6] for the study of free boundary problems without mass conservation in several space dimensions. Let us denote by y = y(ξ, t) the position of the fluid particle which was initially located at the point ξ ∈ [−l, l]. By ρ(y)U (ξ, t) we denote the density at this particle, so that U (ξ, t) = u[y(ξ, t), t]. We consider the fluid motion governed by the following conditions. (1) The trajectory equation: y = v(y, t) ≡ − a(y) ψ (U (ξ, t)) for ξ ∈ [−l, l], t > 0, y t ρ(y) y(ξ, 0) = ξ ∈ [−l, l]. (2) The mass balance law: the mass of every moving fluid volume ω(t) constituted by the same particles changes with time in a prescribed way. Let us take a fluid volume ω(0), label each of the particles by its initial position ξ , and define the mapping ω(0) ∋ ξ → x = y(ξ, t) ∈ ω(t),
t > 0.
The volume ω(t) is constituted by the same particle at each instant t and its mass is given by the formula m(t) = ρ[y(ξ, t)]u[y(ξ, t), t] dy(ξ, t). ω(t)
Let us assume that the mass m(t) of ω(t) changes with time according to the rule dm(t) b(y)h(u) dy. =− dt ω(t) Formally passing to Lagrangian coordinate ξ , we have: dm(t) d = ρ(y)u(y, t)yξ dξ dt dt ω(0) d = ρ(y)u(y, t)yξ dξ = − b(y)h(u)yξ dξ. ω(0) dt ω(0) Since the set ω(0) is arbitrary, we conclude that d ρ(y)u(y, t)yξ = −b(y)h(u)yξ dt
in [−l, l] × [0, T ].
Dividing the both parts of this equality by ρ(y)u(y, t)yξ and integrating in t, we arrive at the mass balance law in Lagrangian setting: t b(y) h(u) ρ(y)u(y, t)yξ = ρ(ξ )u 0 (ξ ) exp − dτ in [−l, l] × [0, T ]. (4.2) 0 ρ(y) u
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It is convenient to introduce now the mass Lagrangian coordinate η, related to the geometrical Lagrangian coordinate ξ by the formula ξ η= ρ(s) u 0 (s)ds − M : [−l, l] → [−M, M]. (4.3) −l
Using the chain rule, it is easy to calculate that 1 1 Dξ = Dη Dξ η yξ yξ t b(y) h(u) ρ(y)u exp dt ρ(ξ )u 0 (ξ )Dη = ρ(ξ )u 0 (ξ ) 0 ρ(y) u = ρ(Y ) A(η, t) U Dη
Dy =
with t
A(η, t) = exp 0
b(Y ) h(U ) dt . ρ(Y ) U
(4.4)
Gathering (4.2) with (4.3) and considering the trajectory y = Y (η, t) and the density u = U (η, t) as functions of the mass coordinate η, we arrive at the following problem: it is requested to find functions (U, Y ) satisfying the system of equations Yt + a(Y ) A U ψη (U ) = 0, ρ(Y ) Yη A U = 1 in Q T = (−M, M) × (0, T ), (4.5) U (−M, t) = U (M, t) = 0, U (η, 0) = u 0 (ξ(η + M)), Y (η, 0) = ξ(η) in [−M, M]. Such a reduction of the free boundary problem (4.1) to problem (4.5) is based on a formal resemblance between the Cauchy problem (4.1) and the mass balance law in the Euler description. Let us now give a rigorous justification of this change of independent variables. Definition 4.1. A pair of functions (Y, U ) is said to be a solution of problem (4.5) in Q T if 1. U, Y ∈ C 0 (Q T ), U ≥ 0 in Q T , √ 2. a(Y )ρ(Y ) U A U ψη (U ) ∈ L 2 (Q T ) and Yt + a(Y )U A ψη (U ) = 0,
ρ(Y ) U A Yη = 1
a.e. in Q T .
(4.6)
Theorem 4.1. Problems (4.1) and (4.5) are equivalent in the following sense: (a) if condition (2.5) is fulfilled, then problem (4.5) has a solution in the sense of Definition 4.1 in the rectangle Q T ∗ (u 0 ) ; (b) if problem (4.5) has a solution in the sense of Definition 4.1 in a rectangle Q T , then problem (4.1) has a solution in the sense of Definition 2.1 in the strip ST and ζ (t) < ∞ on [0, T ]. Proof. (a) Let u(x, t) be a bounded nonnegative weak solution of problem (4.1) in a strip ST ∗ (u 0 ) . Since the data of problem (4.1) are even in x, so is the solution u(x, t). Let us introduce the new independent variable η and the functions Y (η, t), U (η, t) by the relations η Y (η,t) ds = ρ(s)u(s, t) ds, (4.7) 0 0 A(s, t) x = Y (η, t), U (η, t) = u(x, t) with A defined in (4.4). The function Y (η, t) is monotone increasing in η for every fixed t and the transformation η → x = Y (η, t) is locally invertible in a neighborhood of every point where u(x, t) > 0. Denote by ΣT ∗ (u 0 )
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= (−M(t), M(t)) × (0, T ∗ (u 0 )) the image of the set supp u ≡ (−ζ (t), ζ (t)) × (0, T ∗ (u 0 )) under the mapping x → η. By the definition of η and Y (η, t) M(t) ζ (t) ds = ρ(s)u(s, t) ds. A(s, t) 0 0 By the assumption ζ (t) ≤ L on [0, T ∗ (u 0 )]. Choosing in (2.4) a smooth test-function Φ such that 1 if |x| ≤ L + 1, Φ(x, t) = 0 if |x| ≥ L + 2 we arrive at the equality t=T ∗ (u 0 ) =− ρ u d x t=0
R
b(x)h(u(x, t)) d x dt. ST ∗ (u 0 )
The same equality is true for every strip ST with T ≤ T ∗ (u 0 ). It follows that for every t, t + ∆t ∈ [0, T ∗ (u 0 )] τ =t+∆t ζ (τ ) t+∆t ζ (τ ) 1 1 =− ρ u dx b(x)h(u(x, τ )) d x dτ. ∆t 0 ∆t t 0 τ =t
Letting ∆t → 0, we obtain ζ (t) ζ (t) d ρ u dx = − b(x)h(u(x, t)) d x. dt 0 0 The change of variable x = Y (η, t) transforms this equality into the following one: M(t) M(t) M(t) dη d b(Y ) h(u(Y, t)) 1 1 =− dη ≡ dη. dt A(η, t) ρ(Y ) u(Y, t) A A(η, t) t 0 0 0 It follows that M ′ (t) ≡ 0, that is, ΣT ∗ (u 0 ) = Q T ≡ (−M, M) × (0, T ∗ (u 0 )) with l M= ρ u 0 d x. 0
Given a smooth function Φ(x, t), let us define its “double” f (η, t) ≡ Φ(Y (η, t), t) = Φ(x, t)
for (x, t) ∈ supp u and (η, t) ∈ Q T ∗ (u 0 ) .
It is easy to calculate that dη = ρ(Y ) U A dY, ρ(Y ) U A Yη = 1, Dx = ρ(Y ) A U Dη , Φx (x, t)|x=Y (η,t) = ρ(Y ) U A f η , Φt (x, t)|x=Y (η,t) = f t − ρ(Y ) U A Yt f η . With the use of these formulas identity (2.4) can be written in the following way: t=T 0= ρ(x)uΦ d x [ρ(x)u Φt − a(x)Φx u ψx (u) − b(x)h(u)Φ] d xdt − t=0 S R T dηdt = ρ(Y )U f t − ρ(Y )AU Yt f η − a(Y ) f η (ρ(Y )AU )2 U ψη (U ) ρ(Y )AU QT M f (η, t) t=T dηdt dη − − b(Y )h(U ) f t=0 A ρ(Y )AU −M QT M t=T ft f dηdt = dηdt − dη − b(Y )h(U ) f t=0 A A ρ(Y )AU QT −M QT − ρ(Y )U f η Yt + a(Y )AU ψη (U ) dηdt QT
≡ I1 − I2 + I3 + I4 .
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By the definition of A(η, t) f I1 + I3 = dηdt = I2 , A t QT which gives 0 = I4 ≡ − QT
1 Yt + a(Y )AU ψη (U ) dηdt. Φx (x, t) A x=Y (η,t)
Since Φ is arbitrary and A is separated away from zero and infinity, we conclude that the first equation of (4.6) is fulfilled a.e. in Q T . The second equation of (4.6) follows by differentiation of (4.7) with respect to η. Checking of the other properties of U and Y is straightforward. (b) Let the pair (Y, U ) be a solution of problem (4.5) in a rectangle Q T . Introduce the function U (η, t) if x = Y (η, t) with (η, t) ∈ Q T , u(x, t) = 0 otherwise. Since Y ∈ C 0 (Q T ), it is uniformly bounded in Q T . Let Φ(x, t) be an arbitrary test-function satisfying the conditions of Definition 2.1. Set f (η, t) ≡ Φ(Y (η, t), t) = Φ(x, t). Then M Y (M,t) t=T f (η, t) t=T dη = Φ(Y, t)ρ(Y )U (η, t) dY t=0 t=0 A −M Y (−M,t) t=T = Φ(x, t)ρ(x)u(x, t) d x . t=0
R
On the other hand, t M 1 d f (η, t) d b(Y ) h(U ) f (η, t) t=T dη = + f dt dηdt. exp − t=0 A dt dt QT A 0 ρ(Y ) U −M Further, QT
1 d f (η, t) dηdt = A dt
QT
= QT
1 dΦ(Y (η, t), t) dηdt A dt 1 Φt (Y, t) + Φx (Y, t)Yt dηdt A [ρ(Y )U Φt (Y, t) + Φx (Y, t)ρ(Y )U Yt ] dY dt
= QT
= QT
= ST
ρ(Y )U Φt (Y, t) − Φx (Y, t)a(Y )AUρ(Y )U ψη (U ) dY dt
ρ(x)u(x, t)Φt − Φx a(x) φx (u) d xdt
and QT
d f dt
exp − 0
t
b(Y ) h(U ) dt ρ(Y ) U
dηdt = −
f QT
b(Y ) h(U ) dηdt ρ(Y )U A
Φ(x, t)b(x)h(u) d xdt.
=− ST
Comparing these two relations, we obtain the integral identity for the function u(x, t).
5. Auxiliary propositions. The interface equation Proof of Theorem 2.1. The equivalence between problems (4.1) and (4.5) is stated in Theorem 4.1. For every fixed t ∈ [0, T ∗ (u 0 )] the function Y (η, t) is bounded and monotone increasing as a function of η, whence the existence of
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a finite limit ζ (t) ≡ Y (M, t) = lim Y (η, t). η→M−
By virtue of the first equation in (4.5) t Y (η, t) = Y (η, 0) − a(Y )U A ψη (U ) dτ 0 t a(x) ≡ Y (η, 0) − dτ, ψx (u) 0 ρ(x) x=Y (η,τ ) and the assertion follows as η → M−.
Lemma 5.1 (Monotonicity). Let condition (2.3) be fulfilled. Then the solutions of problem (4.1) are monotone decreasing in x ∈ (0, ζ (t)) for every t ∈ (0, T ∗ (u 0 )]. Proof. We argue by contradiction. Let us assume that for some t0 ∈ (0, T ∗ (u 0 )] and some points y, z ∈ (0, ζ (t0 )) (a) 0 < y < z and u(y, t0 ) = α < γ = u(z, t0 ), (b) by continuity of u(x, t), there exist the level curves lα = {(x, t) ∈ ST : u(x, t) = α}
and lγ = {(x, t) ∈ ST : u(x, t) = γ }.
Since the solution u is even, there are only two possibilities. (1) The level curve lγ connects the points (z, t0 ), (−z, t0 ) and does not touch the horizontal line t = 0. Notice that the constant function v = γ is a supersolution of Eq. (4.1). Comparing the functions u(x, t) and v = γ in the domain Dγ , bounded by the curve lγ and the line t = t0 , we arrive at the contradiction with the original hypothesis: u ≥ γ in D γ . (2) The curve lγ continues from the point (z, t0 ) to a point (s, 0). By virtue of the comparison principle the curve lγ can always be chosen in such a way that along lγ the variable t in not increasing from t0 to t = 0. Due to the symmetry of the problem there exists the level curve lγ− which connects the points (−z, t0 ), (−s, 0). Since u 0 is even and monotone decreasing in (0, l), we have that u 0 ≥ γ in [−s, s]. It follows that u ≥ γ in the domain bounded by the curves lγ , lγ− and the straight lines t = 0, t = t0 , which is impossible. Lemma 5.2 (Supersolutions). Let u be a bounded nonnegative solution of problem (4.1) in the strip ST ∗ (u 0 ) . Set µ = ∥u∥∞,ST ∗ (u ) and assume that the following conditions hold: 0 ds h(s) > 0 for s > 0, = ∞. h(s) 0+ If there exists a nonincreasing function λ(x) such that b(x) ≥ λ(x) > 0 ρ(x)
for all x > 0,
then u(x, t) ≤
Hµ−1
t
λ(ζ (τ )) dτ
in ST ,
where
0
Hµ (s) ≡
s
µ
dz . h(z)
(5.1)
Proof. Let us notice first that Hµ (s) : [0, µ] → [0, ∞] is strictly monotone decreasing and has an inverse. Denote by ω(t) the solution of the problem ω′ + λ(ζ (t))h(ω) = 0
for t > 0,
ω(0) = µ,
given by the explicit formula µ t ds Hµ (ω(t)) ≡ = λ(ζ (τ )) dτ. ω(t) h(s) 0
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We have:
1 b(x) Lω ≡ ρ(x) ω (t) − (a(x)φx (ω(t)))x + h(ω(t)) ρ(x) ρ(x) ≥ ρ(x) ω′ (t) + λ(x) h(ω(t)) ≥ ρ(x) ω′ (t) + λ(ζ (t)) h(ω(t)) = 0 ′
in the domain in D = {(x, t) : u(x, t) > 0, t ∈ (0, T ∗ (u 0 )]}, ω(t) > 0 on the boundaries {x = ζ ± (t), t ∈ [0, T ∗ (u 0 ))} of D, and ω(0) = µ ≥ u 0 in (−l, l). By the comparison principle u ≤ ω(t) in D, whence u ≤ ω(t) in ST ∗ (u 0 ) . Corollary 5.1. Let us assume that in the conditions of Lemma 5.2 b(x) ≡ λρ(x) with a positive constant λ. Then the solution of problem (4.1) satisfies the estimate u(x, t) ≤ Hµ−1 (λt)
in ST ∗ (u 0 ) .
(5.2)
The only difference in the proof is that now we take for ω(t) the solution of the problem ω′ (t) + λh(ω(t)) = 0
as t > 0,
ω(0) = µ.
Corollary 5.2. If either h ≡ 0, or b(x) ≡ 0, then estimates (5.1) and (5.2) take on the form: u≤µ
in ST ∗ (u 0 ) .
6. Upper bounds for the interface. Localized solutions 6.1. Proof of Theorem 2.2 (a) Given a solution (Y, U ) of system (4.5), let us write the first equation of (4.5) in the form ρ(Y ) Yt = −ρ(Y )AU ψη (U ), a(Y )
Y (η, 0) ∈ [0, l],
and then integrate it over the interval [0, t]: t t Y (η,t) ρ(s) ∂ ρ(Y ) Yt ds = ds dt a(s) 0 ∂t l 0 a(Y ) l t Y (η,t) ρ(s) ρ(s) = ds − ds = − ρ(Y )A U ψη (U ) dτ. a(s) l Y (η,0) a(s) 0 Choosing z = Y (η, τ ) for the new independent variable and recalling that Y (η, 0) ≤ l, we rewrite this equality as follows: z t l ρ(s) ρ(s) ρ(Y ) A U ψη (U ) η=η(z,τ ) dτ, K = ds ≤ K − ds. (6.1) a(s) 0 0 0 a(s) Integrating (6.1) over the interval (l, ζ (t)) we obtain the inequality ζ (t) z ρ(s) Jl (ζ (t)) ≡ dz ds l l a(s) ζ (t) t ≤ K (ζ (t) − l) − ρ(Y (η, τ ))A U ψη (U ) η=η(z,τ ) dτ dz l
≡ K (ζ (t) − l) −
l
0
ζ (t) t 0
ψx (u(z, τ ))|z=Y (η,τ ) dτ dz.
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Since ζ (τ ) is nondecreasing in τ and u(x, τ ) ≡ 0 for x ≥ Y (M, τ ) ≡ ζ (τ ) by definition, we may rewrite the last inequality in following form: t ζ (τ ) Jl (ζ (t)) ≤ K (ζ (t) − l) − ψx (u) dz dτ 0 l t = K (ζ (t) − l) − [ψ(u(ζ (τ ), τ )) − ψ(u(l, τ ))] dτ 0 t = K (ζ (t) − l) + ψ(u(l, τ )) dτ. 0
Hµ−1 (s) is strictly decreasing, ψ
Recall that is strictly increasing, and λ(θ ) is nonincreasing. Applying (5.1), we arrive at the estimate τ t t λ(ζ (θ )) dθ dτ ψ(u(l, τ )) dτ ≤ ψ Hµ−1 0
0
0
≤
t
ψ
0
Hµ−1 (τ λ(ζ (t)))
1 dτ = λ(ζ (t))
tλ(ζ (t))
0
ψ Hµ−1 (s) ds,
whence R(ζ (t), t) ≤ 0
for all t ≤ T ∗ (u 0 ).
Introducing the new independent variable ξ = Hµ−1 (s), we continue the last inequality as follows: µ t ψ(ξ ) 1 dξ, ψ(u(l, τ )) dτ ≤ −1 λ(ζ (t)) Hµ (tλ(ζ (t))) h(ξ ) 0 (b) If condition (2.8) is fulfilled, then the function λ(s)[Jl (s) − K (s − l)] has an inverse (for large s). Take a sequence {tk } such that tk ↑ T as k → ∞. If T < ∞, then, according to (2.5) and (2.8), |ζ (tk )| ≤ L ,
∥u(·, tk )∥ L ∞ (−L ,L) ≤ ∥u 0 ∥ L ∞ (−l,l)
uniformly with respect to k. It follows that the solution can be continued to a strip R × (T , T + ϵ], which contradicts the definition of T . (c) assume ∞Letus−1 that there exists a sequence {tk } such that ζ (tk ) → ∞ and λ(ζ (tk ))Jl (ζ (tk )) → ∞ as k → ∞. If ψ H ds < ∞, this assumption leads to the contradiction: (s) µ 0 ∞ ∞> ψ Hµ−1 (s) ds ≥ λ(ζ (tk )) [Jl (ζ (tk )) − K (ζ (tk ) − l)] → ∞, 0
which is impossible unless ζ (t) is separated away from infinity.
6.2. Proof of Theorem 2.3 If λ(x) = 0, the upper estimate on Jl (ζ (t)) has the form t Jl (ζ (t)) − K (ζ (t) − l) ≤ ψ(u(l, τ )) dτ ≤ t ψ (µ) . 0
If there exists a sequence {tk } such that Jl (ζ (tk )) → ∞ as k → ∞, then tk ≥
Jl (ζ (tk )) − K (ζ (t) − l) → ∞ as k → ∞. ψ (µ)
7. Lower bounds for the interface Lemma 7.1. Let x0 be the point chosen according to condition (2.10) and D be the constant defined in (2.11). If there exists a positive constant λ such that b(x) = λρ(x), then for every t ∈ [0, T ∗ (u 0 )] the solution of
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problem (4.5) satisfies the inequality ∞ t) d x 1 0 ρ(x)u(x, t) A(x, M ∞ U (M/2, t)A(M/2, t) ≥ ≡ 2 2D ρ(x) d x 0
17
(7.1)
with the constant M from condition (2.3). Proof. We argue by contradiction. Let us assume that for some t0 ∈ [0, T ∗ (u 0 )] and all η ∈ [M/2, M] U (η, t0 , ) A(η, t0 ) < M/2K . According to the second equation of (4.5) (the mass balance law) ∞ ζ (t0 ) Y (M,t0 ) D= ρ(s) ds > ρ(x) d x = ρ(s) ds 0
0 M
= 0
ds ≥ A(s, t0 )U (s, t0 )
0
M M/2
ds M 2D > = D, A(s, t0 )U (s, t0 ) 2 M
which contradicts the assumption. Let us fix an arbitrary t ∈ [0, T ∗ (u 0 )] and choose a point η′ ∈ [M/2, M] such that U (η′ , t)A(η′ , t) ≥ M/2D. Recall that h(U )/U is an increasing function of U , U (η, t) and Y (η, t) are decreasing functions of η, and λ(s) is a decreasing function. Then t t h(U ) h(U ) dτ ≤ exp λ(Y ) dτ = A(M/2, t), A(η′ , t) = exp λ(Y ) U (η′ ,τ ) U (M/2,τ ) 0 0 whence U (M/2, t)A(M/2, t) ≥ U (η′ , t)A(η′ , t) ≥ M/2D. Corollary 7.1. If either h ≡ 0, or b(x) ≡ 0, then estimate (7.1) reads as follows: U (M/2, t) ≥ M/2D for all t ∈ [0, T ∗ (u 0 )]. 7.1. Proof of Theorem 2.4 (a) Let us write the first equation of (4.5) in the form Yt = −ψη (U ) a(Y ) U A and then integrate it over the interval [0, t]: Y (η,t) Y (η,0) t ds ds = − ψη (U ) dτ, a(Y ) U A a(Y ) U A x0 x0 0 where x0 is the point chosen in (2.10). Integration of this equality in the limits [M/2, M] gives M Y (η,t) M Y (η,0) M t ds ds dη − ψη (U ) dτ dη. = a(Y ) U A a(s) u A M/2 x0 M/2 x0 M/2 0 Dropping the nonnegative terms on the right-hand side, we get M Y (η,t) t ds dη ≥ ψ(U (M/2, t)) dτ. a(s) u A M/2 x0 0 t) is monotone decreasing in x, we conclude that Since u A(x, M Y (η,t) t 1 ds I ≡ dη ≥ ψ(U (M/2, τ )) dτ. (η, t), t) x0 a(s) M/2 u(Y (η, t), t) A(Y 0
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Reverting to the variables (x, t), we find that G x0 (ζ (t)) = I . By Lemmas 5.2 and 7.1 the following estimate holds: τ h(U ) M dθ ψ (U (M/2, τ )) ≥ ψ exp − λ(Y ) 2D U (M/2,θ) 0 τ M h(ω) ≥ψ exp − (θ ) dθ λ(Y ) 2D ω 0 τ ′ ω M exp (θ ) dθ =ψ 2D ω 0 M ω(τ ) M =ψ =ψ exp ln ω(τ ) . 2D ω(0) 2µD It follows that G x0 (ζ (t)) ≥
0
t
ψ
M H −1 (λτ ) dτ 2µD µ
=
1 λ
0
λt
ψ
M H −1 (s) ds 2µD µ
= F(t).
(b) Let G x0 (∞) = q < ∞. By the assumption there exists a sequence {tk } such that tk ↗ ∞, ζ (tk ) < ∞ and F(tk ) → ∞ as k → ∞. Along this sequence q ≥ G x0 (ζ (tk )) ≥ F(tk ) → ∞ as k → ∞, which is impossible unless there exists T such that ζ (t) → ∞ as t ↗ T . (c) In this case we apply Corollary 7.1: t M M dτ = t ψ . G x0 (ζ (t)) ≥ ψ 2D 2D 0 (d) In case that G x0 (∞) < ∞, the above inequality is possible only if ζ (t) → ∞ as t ↗ T ∗ .
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